diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1025.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1025.txt new file mode 100644 index 0000000000000000000000000000000000000000..323c57c2f0d87db194c86992766eacb4695c17a4 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1025.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -3 \\ + 8 \\ + 0 \\ + -3 \\ + -7 \\ + -8 \\ + 3 \\ + -3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-3,8,0,-3,-7,-8,3,-3)\, : \\ + \| \, (-3,8,0,-3,-7,-8,3,-3)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-3,8,0,-3,-7,-8,3,-3)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-3,8,0,-3,-7,-8,3,-3)\, \| =\sqrt{(-3)^2+8^2+0^2+(-3)^2+(-7)^2+(-8)^2+3^2+(-3)^2}=\sqrt{213} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1029.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1029.txt new file mode 100644 index 0000000000000000000000000000000000000000..f83c0883c7a48016aa636329f7a05689ec1ca70b --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1029.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{9}{2} \\ + \frac{11}{10} \\ + \frac{39}{10} \\ + -\frac{3}{2} \\ + \frac{39}{10} \\ + -\frac{5}{2} \\ + 5 \\ + \frac{33}{5} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{9}{2},\frac{11}{10},\frac{39}{10},-\frac{3}{2},\frac{39}{10},-\frac{5}{2},5,\frac{33}{5}\right)\, : \\ + \left\| \, \left(-\frac{9}{2},\frac{11}{10},\frac{39}{10},-\frac{3}{2},\frac{39}{10},-\frac{5}{2},5,\frac{33}{5}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{9}{2},\frac{11}{10},\frac{39}{10},-\frac{3}{2},\frac{39}{10},-\frac{5}{2},5,\frac{33}{5}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{9}{2},\frac{11}{10},\frac{39}{10},-\frac{3}{2},\frac{39}{10},-\frac{5}{2},5,\frac{33}{5}\right)\, \right\| =\sqrt{\left(\frac{-9}{2}\right)^2+\left(\frac{11}{10}\right)^2+\left(\frac{39}{10}\right)^2+\left(\frac{-3}{2}\right)^2+\left(\frac{39}{10}\right)^2+\left(\frac{-5}{2}\right)^2+5^2+\left(\frac{33}{5}\right)^2}=\frac{\sqrt{\frac{6447}{2}}}{5} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1188.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1188.txt new file mode 100644 index 0000000000000000000000000000000000000000..1bfd22494ff86e6f58fdacb0f2f6bc2d4d8a3767 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1188.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 8 \\ + \frac{7}{2} \\ + 9 \\ + 8 \\ + 1 \\ + -5 \\ + \frac{3}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(8,\frac{7}{2},9,8,1,-5,\frac{3}{2}\right)\, : \\ + \left\| \, \left(8,\frac{7}{2},9,8,1,-5,\frac{3}{2}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(8,\frac{7}{2},9,8,1,-5,\frac{3}{2}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(8,\frac{7}{2},9,8,1,-5,\frac{3}{2}\right)\, \right\| =\sqrt{8^2+\left(\frac{7}{2}\right)^2+9^2+8^2+1^2+(-5)^2+\left(\frac{3}{2}\right)^2}=\sqrt{\frac{499}{2}} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1218.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1218.txt new file mode 100644 index 0000000000000000000000000000000000000000..ed69987dffdc3356bf1f4a857918d013d77747a3 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1218.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 10 \\ + -6 \\ + -4 \\ + 8 \\ + 7 \\ + -5 \\ + 5 \\ + 8 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (10,-6,-4,8,7,-5,5,8)\, : \\ + \| \, (10,-6,-4,8,7,-5,5,8)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (10,-6,-4,8,7,-5,5,8)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (10,-6,-4,8,7,-5,5,8)\, \| =\sqrt{10^2+(-6)^2+(-4)^2+8^2+7^2+(-5)^2+5^2+8^2}=\sqrt{379} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1387.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1387.txt new file mode 100644 index 0000000000000000000000000000000000000000..62776f8bf2bb1a6f4954c7c5e415c945aa1bec4a --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1387.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -1 \\ + -4 \\ + 0 \\ + -6 \\ + -7 \\ + 3 \\ + 3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-1,-4,0,-6,-7,3,3)\, : \\ + \| \, (-1,-4,0,-6,-7,3,3)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-1,-4,0,-6,-7,3,3)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-1,-4,0,-6,-7,3,3)\, \| =\sqrt{(-1)^2+(-4)^2+0^2+(-6)^2+(-7)^2+3^2+3^2}=2 \sqrt{30} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1428.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1428.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d38329a6411bdd4524ffcc882355834277b768b --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1428.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -9 \\ + -4 \\ + -1 \\ + 1 \\ + -10 \\ + 9 \\ + -7 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-9,-4,-1,1,-10,9,-7)\, : \\ + \| \, (-9,-4,-1,1,-10,9,-7)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-9,-4,-1,1,-10,9,-7)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-9,-4,-1,1,-10,9,-7)\, \| =\sqrt{(-9)^2+(-4)^2+(-1)^2+1^2+(-10)^2+9^2+(-7)^2}=\sqrt{329} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/149.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/149.txt new file mode 100644 index 0000000000000000000000000000000000000000..70288888008ca5e3f6b107734d537281e9dcc0b0 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/149.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -7 \\ + 3 \\ + 3 \\ + 1 \\ + 5 \\ + 2 \\ + 2 \\ + -9 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-7,3,3,1,5,2,2,-9)\, : \\ + \| \, (-7,3,3,1,5,2,2,-9)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-7,3,3,1,5,2,2,-9)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-7,3,3,1,5,2,2,-9)\, \| =\sqrt{(-7)^2+3^2+3^2+1^2+5^2+2^2+2^2+(-9)^2}=\sqrt{182} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1510.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1510.txt new file mode 100644 index 0000000000000000000000000000000000000000..273c439540868728cabae2ade06006ecb1329ff9 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1510.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -1 \\ + -3 \\ + 4 \\ + -5 \\ + -3 \\ + -4 \\ + 2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-1,-3,4,-5,-3,-4,2)\, : \\ + \| \, (-1,-3,4,-5,-3,-4,2)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-1,-3,4,-5,-3,-4,2)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-1,-3,4,-5,-3,-4,2)\, \| =\sqrt{(-1)^2+(-3)^2+4^2+(-5)^2+(-3)^2+(-4)^2+2^2}=4 \sqrt{5} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1516.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1516.txt new file mode 100644 index 0000000000000000000000000000000000000000..a475af9831a81eea294089e57a0c571d867876ef --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1516.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{49}{5} \\ + \frac{4}{5} \\ + 4 \\ + -\frac{67}{10} \\ + \frac{63}{10} \\ + -\frac{49}{5} \\ + \frac{14}{5} \\ + \frac{16}{5} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{49}{5},\frac{4}{5},4,-\frac{67}{10},\frac{63}{10},-\frac{49}{5},\frac{14}{5},\frac{16}{5}\right)\, : \\ + \left\| \, \left(-\frac{49}{5},\frac{4}{5},4,-\frac{67}{10},\frac{63}{10},-\frac{49}{5},\frac{14}{5},\frac{16}{5}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{49}{5},\frac{4}{5},4,-\frac{67}{10},\frac{63}{10},-\frac{49}{5},\frac{14}{5},\frac{16}{5}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{49}{5},\frac{4}{5},4,-\frac{67}{10},\frac{63}{10},-\frac{49}{5},\frac{14}{5},\frac{16}{5}\right)\, \right\| =\sqrt{\left(\frac{-49}{5}\right)^2+\left(\frac{4}{5}\right)^2+4^2+\left(\frac{-67}{10}\right)^2+\left(\frac{63}{10}\right)^2+\left(\frac{-49}{5}\right)^2+\left(\frac{14}{5}\right)^2+\left(\frac{16}{5}\right)^2}=\frac{\sqrt{\frac{15569}{2}}}{5} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1545.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1545.txt new file mode 100644 index 0000000000000000000000000000000000000000..dbaaa97f0032a1e5352caf7f6d8a0c89edb95c03 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1545.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{5}{4} \\ + \frac{17}{4} \\ + \frac{15}{4} \\ + 6 \\ + \frac{5}{4} \\ + -\frac{5}{4} \\ + 6 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{5}{4},\frac{17}{4},\frac{15}{4},6,\frac{5}{4},-\frac{5}{4},6\right)\, : \\ + \left\| \, \left(\frac{5}{4},\frac{17}{4},\frac{15}{4},6,\frac{5}{4},-\frac{5}{4},6\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{5}{4},\frac{17}{4},\frac{15}{4},6,\frac{5}{4},-\frac{5}{4},6\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{5}{4},\frac{17}{4},\frac{15}{4},6,\frac{5}{4},-\frac{5}{4},6\right)\, \right\| =\sqrt{\left(\frac{5}{4}\right)^2+\left(\frac{17}{4}\right)^2+\left(\frac{15}{4}\right)^2+6^2+\left(\frac{5}{4}\right)^2+\left(\frac{-5}{4}\right)^2+6^2}=\frac{\sqrt{1741}}{4} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1574.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1574.txt new file mode 100644 index 0000000000000000000000000000000000000000..876f3777b176edd9a80e5e452d60830c569600e7 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1574.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 5 \\ + 9 \\ + 2 \\ + 1 \\ + 1 \\ + 2 \\ + 5 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (5,9,2,1,1,2,5)\, : \\ + \| \, (5,9,2,1,1,2,5)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (5,9,2,1,1,2,5)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (5,9,2,1,1,2,5)\, \| =\sqrt{5^2+9^2+2^2+1^2+1^2+2^2+5^2}=\sqrt{141} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/168.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/168.txt new file mode 100644 index 0000000000000000000000000000000000000000..a085dd609854cd80f02b3e57124f55cbffc6b71c --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/168.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 7 \\ + -5 \\ + -4 \\ + 10 \\ + -1 \\ + -6 \\ + -8 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (7,-5,-4,10,-1,-6,-8)\, : \\ + \| \, (7,-5,-4,10,-1,-6,-8)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (7,-5,-4,10,-1,-6,-8)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (7,-5,-4,10,-1,-6,-8)\, \| =\sqrt{7^2+(-5)^2+(-4)^2+10^2+(-1)^2+(-6)^2+(-8)^2}=\sqrt{291} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1737.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1737.txt new file mode 100644 index 0000000000000000000000000000000000000000..305de75fab4ab59aa6a08ce094f2e64c1898fae1 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1737.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -8 \\ + -\frac{44}{7} \\ + -\frac{38}{7} \\ + -\frac{50}{7} \\ + -\frac{17}{7} \\ + \frac{38}{7} \\ + -7 \\ + -\frac{8}{7} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-8,-\frac{44}{7},-\frac{38}{7},-\frac{50}{7},-\frac{17}{7},\frac{38}{7},-7,-\frac{8}{7}\right)\, : \\ + \left\| \, \left(-8,-\frac{44}{7},-\frac{38}{7},-\frac{50}{7},-\frac{17}{7},\frac{38}{7},-7,-\frac{8}{7}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-8,-\frac{44}{7},-\frac{38}{7},-\frac{50}{7},-\frac{17}{7},\frac{38}{7},-7,-\frac{8}{7}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-8,-\frac{44}{7},-\frac{38}{7},-\frac{50}{7},-\frac{17}{7},\frac{38}{7},-7,-\frac{8}{7}\right)\, \right\| =\sqrt{(-8)^2+\left(\frac{-44}{7}\right)^2+\left(\frac{-38}{7}\right)^2+\left(\frac{-50}{7}\right)^2+\left(\frac{-17}{7}\right)^2+\left(\frac{38}{7}\right)^2+(-7)^2+\left(\frac{-8}{7}\right)^2}=\frac{\sqrt{13214}}{7} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1741.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1741.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a1f5bff04189fb9aa117887f1522db642761ca5 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1741.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{83}{16} \\ + -\frac{25}{8} \\ + \frac{59}{8} \\ + -\frac{23}{8} \\ + -\frac{43}{8} \\ + -\frac{55}{16} \\ + \frac{13}{16} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{83}{16},-\frac{25}{8},\frac{59}{8},-\frac{23}{8},-\frac{43}{8},-\frac{55}{16},\frac{13}{16}\right)\, : \\ + \left\| \, \left(-\frac{83}{16},-\frac{25}{8},\frac{59}{8},-\frac{23}{8},-\frac{43}{8},-\frac{55}{16},\frac{13}{16}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{83}{16},-\frac{25}{8},\frac{59}{8},-\frac{23}{8},-\frac{43}{8},-\frac{55}{16},\frac{13}{16}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{83}{16},-\frac{25}{8},\frac{59}{8},-\frac{23}{8},-\frac{43}{8},-\frac{55}{16},\frac{13}{16}\right)\, \right\| =\sqrt{\left(\frac{-83}{16}\right)^2+\left(\frac{-25}{8}\right)^2+\left(\frac{59}{8}\right)^2+\left(\frac{-23}{8}\right)^2+\left(\frac{-43}{8}\right)^2+\left(\frac{-55}{16}\right)^2+\left(\frac{13}{16}\right)^2}=\frac{\sqrt{36019}}{16} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1791.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1791.txt new file mode 100644 index 0000000000000000000000000000000000000000..b12c9c4026936ee13b891240127ae1cd71989e97 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1791.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 7 \\ + -\frac{121}{16} \\ + -\frac{91}{16} \\ + \frac{51}{8} \\ + 3 \\ + -\frac{41}{8} \\ + -\frac{131}{16} \\ + -\frac{15}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(7,-\frac{121}{16},-\frac{91}{16},\frac{51}{8},3,-\frac{41}{8},-\frac{131}{16},-\frac{15}{2}\right)\, : \\ + \left\| \, \left(7,-\frac{121}{16},-\frac{91}{16},\frac{51}{8},3,-\frac{41}{8},-\frac{131}{16},-\frac{15}{2}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(7,-\frac{121}{16},-\frac{91}{16},\frac{51}{8},3,-\frac{41}{8},-\frac{131}{16},-\frac{15}{2}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(7,-\frac{121}{16},-\frac{91}{16},\frac{51}{8},3,-\frac{41}{8},-\frac{131}{16},-\frac{15}{2}\right)\, \right\| =\sqrt{7^2+\left(\frac{-121}{16}\right)^2+\left(\frac{-91}{16}\right)^2+\left(\frac{51}{8}\right)^2+3^2+\left(\frac{-41}{8}\right)^2+\left(\frac{-131}{16}\right)^2+\left(\frac{-15}{2}\right)^2}=\frac{\sqrt{86459}}{16} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/18.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/18.txt new file mode 100644 index 0000000000000000000000000000000000000000..558c67cf61a0764fa98e67b410c856ee2cc9058a --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/18.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 1 \\ + 9 \\ + -4 \\ + 1 \\ + 7 \\ + -4 \\ + -2 \\ + 5 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (1,9,-4,1,7,-4,-2,5)\, : \\ + \| \, (1,9,-4,1,7,-4,-2,5)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (1,9,-4,1,7,-4,-2,5)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (1,9,-4,1,7,-4,-2,5)\, \| =\sqrt{1^2+9^2+(-4)^2+1^2+7^2+(-4)^2+(-2)^2+5^2}=\sqrt{193} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1944.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1944.txt new file mode 100644 index 0000000000000000000000000000000000000000..2a3e6d4ab82860bc07a13eb15c4c6a9d4876f584 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1944.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -1 \\ + 5 \\ + -7 \\ + 4 \\ + 5 \\ + -6 \\ + -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-1,5,-7,4,5,-6,-2)\, : \\ + \| \, (-1,5,-7,4,5,-6,-2)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-1,5,-7,4,5,-6,-2)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-1,5,-7,4,5,-6,-2)\, \| =\sqrt{(-1)^2+5^2+(-7)^2+4^2+5^2+(-6)^2+(-2)^2}=2 \sqrt{39} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1949.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1949.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad6e482ab742c1756eb5f5abba976068f57043dc --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1949.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{7}{4} \\ + \frac{23}{4} \\ + \frac{13}{4} \\ + \frac{35}{4} \\ + 3 \\ + \frac{25}{4} \\ + 5 \\ + \frac{37}{4} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{7}{4},\frac{23}{4},\frac{13}{4},\frac{35}{4},3,\frac{25}{4},5,\frac{37}{4}\right)\, : \\ + \left\| \, \left(-\frac{7}{4},\frac{23}{4},\frac{13}{4},\frac{35}{4},3,\frac{25}{4},5,\frac{37}{4}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{7}{4},\frac{23}{4},\frac{13}{4},\frac{35}{4},3,\frac{25}{4},5,\frac{37}{4}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{7}{4},\frac{23}{4},\frac{13}{4},\frac{35}{4},3,\frac{25}{4},5,\frac{37}{4}\right)\, \right\| =\sqrt{\left(\frac{-7}{4}\right)^2+\left(\frac{23}{4}\right)^2+\left(\frac{13}{4}\right)^2+\left(\frac{35}{4}\right)^2+3^2+\left(\frac{25}{4}\right)^2+5^2+\left(\frac{37}{4}\right)^2}=\frac{\sqrt{\frac{2255}{2}}}{2} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1959.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1959.txt new file mode 100644 index 0000000000000000000000000000000000000000..b6f6b09c20e2782d3e35d89acba2bfe0a42ae491 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/1959.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -5 \\ + -\frac{3}{2} \\ + -6 \\ + -\frac{3}{2} \\ + -4 \\ + 4 \\ + \frac{13}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-5,-\frac{3}{2},-6,-\frac{3}{2},-4,4,\frac{13}{2}\right)\, : \\ + \left\| \, \left(-5,-\frac{3}{2},-6,-\frac{3}{2},-4,4,\frac{13}{2}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-5,-\frac{3}{2},-6,-\frac{3}{2},-4,4,\frac{13}{2}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-5,-\frac{3}{2},-6,-\frac{3}{2},-4,4,\frac{13}{2}\right)\, \right\| =\sqrt{(-5)^2+\left(\frac{-3}{2}\right)^2+(-6)^2+\left(\frac{-3}{2}\right)^2+(-4)^2+4^2+\left(\frac{13}{2}\right)^2}=\frac{\sqrt{559}}{2} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/202.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/202.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac662c75bdfae9ab61cf2ba0ed14c5d1ab6aec51 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/202.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{96}{25} \\ + \frac{531}{100} \\ + \frac{19}{5} \\ + -\frac{437}{50} \\ + -\frac{59}{100} \\ + \frac{141}{50} \\ + -\frac{473}{50} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{96}{25},\frac{531}{100},\frac{19}{5},-\frac{437}{50},-\frac{59}{100},\frac{141}{50},-\frac{473}{50}\right)\, : \\ + \left\| \, \left(-\frac{96}{25},\frac{531}{100},\frac{19}{5},-\frac{437}{50},-\frac{59}{100},\frac{141}{50},-\frac{473}{50}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{96}{25},\frac{531}{100},\frac{19}{5},-\frac{437}{50},-\frac{59}{100},\frac{141}{50},-\frac{473}{50}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{96}{25},\frac{531}{100},\frac{19}{5},-\frac{437}{50},-\frac{59}{100},\frac{141}{50},-\frac{473}{50}\right)\, \right\| =\sqrt{\left(\frac{-96}{25}\right)^2+\left(\frac{531}{100}\right)^2+\left(\frac{19}{5}\right)^2+\left(\frac{-437}{50}\right)^2+\left(\frac{-59}{100}\right)^2+\left(\frac{141}{50}\right)^2+\left(\frac{-473}{50}\right)^2}=\frac{\sqrt{\frac{1157807}{2}}}{50} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2094.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2094.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d6a5333b570d3876a43fdf645257a7289fc4e27 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2094.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -8 \\ + 8 \\ + 0 \\ + 4 \\ + 8 \\ + 2 \\ + -1 \\ + -6 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-8,8,0,4,8,2,-1,-6)\, : \\ + \| \, (-8,8,0,4,8,2,-1,-6)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-8,8,0,4,8,2,-1,-6)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-8,8,0,4,8,2,-1,-6)\, \| =\sqrt{(-8)^2+8^2+0^2+4^2+8^2+2^2+(-1)^2+(-6)^2}=\sqrt{249} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/212.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/212.txt new file mode 100644 index 0000000000000000000000000000000000000000..ba731647cd67016c4d046a2bf81af4b56b097caf --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/212.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{643}{100} \\ + \frac{99}{25} \\ + -\frac{273}{100} \\ + -\frac{989}{100} \\ + \frac{127}{25} \\ + -\frac{68}{25} \\ + -\frac{561}{100} \\ + \frac{229}{25} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{643}{100},\frac{99}{25},-\frac{273}{100},-\frac{989}{100},\frac{127}{25},-\frac{68}{25},-\frac{561}{100},\frac{229}{25}\right)\, : \\ + \left\| \, \left(-\frac{643}{100},\frac{99}{25},-\frac{273}{100},-\frac{989}{100},\frac{127}{25},-\frac{68}{25},-\frac{561}{100},\frac{229}{25}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{643}{100},\frac{99}{25},-\frac{273}{100},-\frac{989}{100},\frac{127}{25},-\frac{68}{25},-\frac{561}{100},\frac{229}{25}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{643}{100},\frac{99}{25},-\frac{273}{100},-\frac{989}{100},\frac{127}{25},-\frac{68}{25},-\frac{561}{100},\frac{229}{25}\right)\, \right\| =\sqrt{\left(\frac{-643}{100}\right)^2+\left(\frac{99}{25}\right)^2+\left(\frac{-273}{100}\right)^2+\left(\frac{-989}{100}\right)^2+\left(\frac{127}{25}\right)^2+\left(\frac{-68}{25}\right)^2+\left(\frac{-561}{100}\right)^2+\left(\frac{229}{25}\right)^2}=\frac{\sqrt{\frac{155437}{5}}}{10} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2198.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2198.txt new file mode 100644 index 0000000000000000000000000000000000000000..35e2a6288f57028b38d0cb8f5535cfd7e0de770f --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2198.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -4 \\ + 8 \\ + 2 \\ + -5 \\ + -6 \\ + 1 \\ + -4 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-4,8,2,-5,-6,1,-4)\, : \\ + \| \, (-4,8,2,-5,-6,1,-4)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-4,8,2,-5,-6,1,-4)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-4,8,2,-5,-6,1,-4)\, \| =\sqrt{(-4)^2+8^2+2^2+(-5)^2+(-6)^2+1^2+(-4)^2}=9 \sqrt{2} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2236.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2236.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b345390922f134672a2d1349018a577a0becf94 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2236.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{8}{3} \\ + -\frac{20}{3} \\ + \frac{41}{6} \\ + \frac{3}{2} \\ + -\frac{1}{3} \\ + -\frac{26}{3} \\ + -\frac{13}{3} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{8}{3},-\frac{20}{3},\frac{41}{6},\frac{3}{2},-\frac{1}{3},-\frac{26}{3},-\frac{13}{3}\right)\, : \\ + \left\| \, \left(\frac{8}{3},-\frac{20}{3},\frac{41}{6},\frac{3}{2},-\frac{1}{3},-\frac{26}{3},-\frac{13}{3}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{8}{3},-\frac{20}{3},\frac{41}{6},\frac{3}{2},-\frac{1}{3},-\frac{26}{3},-\frac{13}{3}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{8}{3},-\frac{20}{3},\frac{41}{6},\frac{3}{2},-\frac{1}{3},-\frac{26}{3},-\frac{13}{3}\right)\, \right\| =\sqrt{\left(\frac{8}{3}\right)^2+\left(\frac{-20}{3}\right)^2+\left(\frac{41}{6}\right)^2+\left(\frac{3}{2}\right)^2+\left(\frac{-1}{3}\right)^2+\left(\frac{-26}{3}\right)^2+\left(\frac{-13}{3}\right)^2}=\sqrt{\frac{389}{2}} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2246.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2246.txt new file mode 100644 index 0000000000000000000000000000000000000000..4798879d320756ac1c04fde371fbe7402d74fef6 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2246.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 10 \\ + 2 \\ + 9 \\ + -8 \\ + -2 \\ + -6 \\ + 0 \\ + -5 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (10,2,9,-8,-2,-6,0,-5)\, : \\ + \| \, (10,2,9,-8,-2,-6,0,-5)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (10,2,9,-8,-2,-6,0,-5)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (10,2,9,-8,-2,-6,0,-5)\, \| =\sqrt{10^2+2^2+9^2+(-8)^2+(-2)^2+(-6)^2+0^2+(-5)^2}=\sqrt{314} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2336.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2336.txt new file mode 100644 index 0000000000000000000000000000000000000000..0305ec62ae9611ee129ec5363f263f1ad85979d6 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2336.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 2 \\ + -1 \\ + 5 \\ + -9 \\ + \frac{15}{2} \\ + \frac{15}{2} \\ + \frac{17}{2} \\ + -7 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(2,-1,5,-9,\frac{15}{2},\frac{15}{2},\frac{17}{2},-7\right)\, : \\ + \left\| \, \left(2,-1,5,-9,\frac{15}{2},\frac{15}{2},\frac{17}{2},-7\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(2,-1,5,-9,\frac{15}{2},\frac{15}{2},\frac{17}{2},-7\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(2,-1,5,-9,\frac{15}{2},\frac{15}{2},\frac{17}{2},-7\right)\, \right\| =\sqrt{2^2+(-1)^2+5^2+(-9)^2+\left(\frac{15}{2}\right)^2+\left(\frac{15}{2}\right)^2+\left(\frac{17}{2}\right)^2+(-7)^2}=\frac{\sqrt{1379}}{2} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2414.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2414.txt new file mode 100644 index 0000000000000000000000000000000000000000..4468d118d864f88e5d27e27293e85e8e34cbaca2 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2414.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 5 \\ + 10 \\ + 10 \\ + 6 \\ + 1 \\ + 5 \\ + -8 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (5,10,10,6,1,5,-8)\, : \\ + \| \, (5,10,10,6,1,5,-8)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (5,10,10,6,1,5,-8)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (5,10,10,6,1,5,-8)\, \| =\sqrt{5^2+10^2+10^2+6^2+1^2+5^2+(-8)^2}=3 \sqrt{39} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2430.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2430.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e6aab2eac7707ecd11979b2bb5c8d060d2ae5ca --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2430.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -8 \\ + -7 \\ + 4 \\ + -5 \\ + -3 \\ + -2 \\ + 2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-8,-7,4,-5,-3,-2,2)\, : \\ + \| \, (-8,-7,4,-5,-3,-2,2)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-8,-7,4,-5,-3,-2,2)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-8,-7,4,-5,-3,-2,2)\, \| =\sqrt{(-8)^2+(-7)^2+4^2+(-5)^2+(-3)^2+(-2)^2+2^2}=3 \sqrt{19} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2436.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2436.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1fc338f9c92afec2a2702a84f8623287ae928ca --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2436.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{19}{2} \\ + \frac{39}{4} \\ + 1 \\ + 6 \\ + -5 \\ + -6 \\ + \frac{11}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{19}{2},\frac{39}{4},1,6,-5,-6,\frac{11}{2}\right)\, : \\ + \left\| \, \left(\frac{19}{2},\frac{39}{4},1,6,-5,-6,\frac{11}{2}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{19}{2},\frac{39}{4},1,6,-5,-6,\frac{11}{2}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{19}{2},\frac{39}{4},1,6,-5,-6,\frac{11}{2}\right)\, \right\| =\sqrt{\left(\frac{19}{2}\right)^2+\left(\frac{39}{4}\right)^2+1^2+6^2+(-5)^2+(-6)^2+\left(\frac{11}{2}\right)^2}=\frac{\sqrt{5017}}{4} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2723.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2723.txt new file mode 100644 index 0000000000000000000000000000000000000000..52087be9fc5fb3423f75d6f1db9a44d39932fe63 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2723.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -3 \\ + -9 \\ + 1 \\ + 2 \\ + -3 \\ + 9 \\ + 3 \\ + -6 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-3,-9,1,2,-3,9,3,-6)\, : \\ + \| \, (-3,-9,1,2,-3,9,3,-6)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-3,-9,1,2,-3,9,3,-6)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-3,-9,1,2,-3,9,3,-6)\, \| =\sqrt{(-3)^2+(-9)^2+1^2+2^2+(-3)^2+9^2+3^2+(-6)^2}=\sqrt{230} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2997.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2997.txt new file mode 100644 index 0000000000000000000000000000000000000000..a78ed5ab8f8b495ac3288b80e3e4214bc24dd176 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/2997.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{17}{2} \\ + -\frac{11}{2} \\ + \frac{3}{2} \\ + -3 \\ + -5 \\ + -10 \\ + 6 \\ + -\frac{13}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{17}{2},-\frac{11}{2},\frac{3}{2},-3,-5,-10,6,-\frac{13}{2}\right)\, : \\ + \left\| \, \left(\frac{17}{2},-\frac{11}{2},\frac{3}{2},-3,-5,-10,6,-\frac{13}{2}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{17}{2},-\frac{11}{2},\frac{3}{2},-3,-5,-10,6,-\frac{13}{2}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{17}{2},-\frac{11}{2},\frac{3}{2},-3,-5,-10,6,-\frac{13}{2}\right)\, \right\| =\sqrt{\left(\frac{17}{2}\right)^2+\left(\frac{-11}{2}\right)^2+\left(\frac{3}{2}\right)^2+(-3)^2+(-5)^2+(-10)^2+6^2+\left(\frac{-13}{2}\right)^2}=\sqrt{317} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3074.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3074.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc8f63d81b7c39dd9082bc836e181d1cffdb0dbb --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3074.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -9 \\ + -3 \\ + 7 \\ + 6 \\ + -2 \\ + -2 \\ + 2 \\ + -8 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-9,-3,7,6,-2,-2,2,-8)\, : \\ + \| \, (-9,-3,7,6,-2,-2,2,-8)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-9,-3,7,6,-2,-2,2,-8)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-9,-3,7,6,-2,-2,2,-8)\, \| =\sqrt{(-9)^2+(-3)^2+7^2+6^2+(-2)^2+(-2)^2+2^2+(-8)^2}=\sqrt{251} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3083.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3083.txt new file mode 100644 index 0000000000000000000000000000000000000000..dae0aae5e558806a8ad77967732778d640efb30a --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3083.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{101}{16} \\ + -1 \\ + -\frac{1}{16} \\ + -\frac{87}{16} \\ + -\frac{21}{8} \\ + \frac{147}{16} \\ + \frac{23}{8} \\ + -1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{101}{16},-1,-\frac{1}{16},-\frac{87}{16},-\frac{21}{8},\frac{147}{16},\frac{23}{8},-1\right)\, : \\ + \left\| \, \left(-\frac{101}{16},-1,-\frac{1}{16},-\frac{87}{16},-\frac{21}{8},\frac{147}{16},\frac{23}{8},-1\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{101}{16},-1,-\frac{1}{16},-\frac{87}{16},-\frac{21}{8},\frac{147}{16},\frac{23}{8},-1\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{101}{16},-1,-\frac{1}{16},-\frac{87}{16},-\frac{21}{8},\frac{147}{16},\frac{23}{8},-1\right)\, \right\| =\sqrt{\left(\frac{-101}{16}\right)^2+(-1)^2+\left(\frac{-1}{16}\right)^2+\left(\frac{-87}{16}\right)^2+\left(\frac{-21}{8}\right)^2+\left(\frac{147}{16}\right)^2+\left(\frac{23}{8}\right)^2+(-1)^2}=\frac{\sqrt{10943}}{8} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3091.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3091.txt new file mode 100644 index 0000000000000000000000000000000000000000..66dfbe82128f72fda36bdb3ec0a5b133b2ec2824 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3091.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -7 \\ + -7 \\ + 10 \\ + 6 \\ + -5 \\ + -3 \\ + 9 \\ + 5 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-7,-7,10,6,-5,-3,9,5)\, : \\ + \| \, (-7,-7,10,6,-5,-3,9,5)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-7,-7,10,6,-5,-3,9,5)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-7,-7,10,6,-5,-3,9,5)\, \| =\sqrt{(-7)^2+(-7)^2+10^2+6^2+(-5)^2+(-3)^2+9^2+5^2}=\sqrt{374} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3290.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3290.txt new file mode 100644 index 0000000000000000000000000000000000000000..ba512b7663e552dad654a1f3b9420f293c1a20e6 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3290.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{2}{5} \\ + -6 \\ + \frac{19}{5} \\ + -\frac{21}{5} \\ + \frac{17}{5} \\ + \frac{4}{5} \\ + -\frac{22}{5} \\ + -\frac{26}{5} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{2}{5},-6,\frac{19}{5},-\frac{21}{5},\frac{17}{5},\frac{4}{5},-\frac{22}{5},-\frac{26}{5}\right)\, : \\ + \left\| \, \left(\frac{2}{5},-6,\frac{19}{5},-\frac{21}{5},\frac{17}{5},\frac{4}{5},-\frac{22}{5},-\frac{26}{5}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{2}{5},-6,\frac{19}{5},-\frac{21}{5},\frac{17}{5},\frac{4}{5},-\frac{22}{5},-\frac{26}{5}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{2}{5},-6,\frac{19}{5},-\frac{21}{5},\frac{17}{5},\frac{4}{5},-\frac{22}{5},-\frac{26}{5}\right)\, \right\| =\sqrt{\left(\frac{2}{5}\right)^2+(-6)^2+\left(\frac{19}{5}\right)^2+\left(\frac{-21}{5}\right)^2+\left(\frac{17}{5}\right)^2+\left(\frac{4}{5}\right)^2+\left(\frac{-22}{5}\right)^2+\left(\frac{-26}{5}\right)^2}=\frac{\sqrt{3171}}{5} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3341.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3341.txt new file mode 100644 index 0000000000000000000000000000000000000000..018f47dee714027254f53028d4b01f9b239f8f25 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3341.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -7 \\ + -7 \\ + 8 \\ + 9 \\ + -8 \\ + 5 \\ + 10 \\ + -1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-7,-7,8,9,-8,5,10,-1)\, : \\ + \| \, (-7,-7,8,9,-8,5,10,-1)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-7,-7,8,9,-8,5,10,-1)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-7,-7,8,9,-8,5,10,-1)\, \| =\sqrt{(-7)^2+(-7)^2+8^2+9^2+(-8)^2+5^2+10^2+(-1)^2}=\sqrt{433} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3409.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3409.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbd4978d09dd10b711f440f6ebda0daf84b934b1 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3409.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{45}{8} \\ + \frac{35}{16} \\ + \frac{95}{16} \\ + \frac{117}{16} \\ + \frac{73}{8} \\ + \frac{43}{16} \\ + \frac{159}{16} \\ + -\frac{37}{16} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{45}{8},\frac{35}{16},\frac{95}{16},\frac{117}{16},\frac{73}{8},\frac{43}{16},\frac{159}{16},-\frac{37}{16}\right)\, : \\ + \left\| \, \left(-\frac{45}{8},\frac{35}{16},\frac{95}{16},\frac{117}{16},\frac{73}{8},\frac{43}{16},\frac{159}{16},-\frac{37}{16}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{45}{8},\frac{35}{16},\frac{95}{16},\frac{117}{16},\frac{73}{8},\frac{43}{16},\frac{159}{16},-\frac{37}{16}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{45}{8},\frac{35}{16},\frac{95}{16},\frac{117}{16},\frac{73}{8},\frac{43}{16},\frac{159}{16},-\frac{37}{16}\right)\, \right\| =\sqrt{\left(\frac{-45}{8}\right)^2+\left(\frac{35}{16}\right)^2+\left(\frac{95}{16}\right)^2+\left(\frac{117}{16}\right)^2+\left(\frac{73}{8}\right)^2+\left(\frac{43}{16}\right)^2+\left(\frac{159}{16}\right)^2+\left(\frac{-37}{16}\right)^2}=\frac{\sqrt{\frac{40927}{2}}}{8} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/341.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/341.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d6a98c2abb91c28f0ff87da2858f7e437862861 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/341.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -4 \\ + 4 \\ + 7 \\ + -5 \\ + -7 \\ + -10 \\ + -8 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-4,4,7,-5,-7,-10,-8)\, : \\ + \| \, (-4,4,7,-5,-7,-10,-8)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-4,4,7,-5,-7,-10,-8)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-4,4,7,-5,-7,-10,-8)\, \| =\sqrt{(-4)^2+4^2+7^2+(-5)^2+(-7)^2+(-10)^2+(-8)^2}=\sqrt{319} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/344.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/344.txt new file mode 100644 index 0000000000000000000000000000000000000000..28986b474429ebacba8b061e69af63049f2a065d --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/344.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -5 \\ + -8 \\ + 4 \\ + -8 \\ + 1 \\ + -2 \\ + 9 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-5,-8,4,-8,1,-2,9)\, : \\ + \| \, (-5,-8,4,-8,1,-2,9)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-5,-8,4,-8,1,-2,9)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-5,-8,4,-8,1,-2,9)\, \| =\sqrt{(-5)^2+(-8)^2+4^2+(-8)^2+1^2+(-2)^2+9^2}=\sqrt{255} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/345.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/345.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca60a69417439bc2d02c4a47a0648aaa8e2f9174 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/345.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 4 \\ + 9 \\ + -2 \\ + 6 \\ + 7 \\ + 1 \\ + -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (4,9,-2,6,7,1,-2)\, : \\ + \| \, (4,9,-2,6,7,1,-2)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (4,9,-2,6,7,1,-2)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (4,9,-2,6,7,1,-2)\, \| =\sqrt{4^2+9^2+(-2)^2+6^2+7^2+1^2+(-2)^2}=\sqrt{191} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3502.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3502.txt new file mode 100644 index 0000000000000000000000000000000000000000..b6266f128b2984c10f6a2529ed9de35a75af7959 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3502.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{13}{2} \\ + \frac{5}{2} \\ + \frac{13}{2} \\ + -2 \\ + 10 \\ + \frac{3}{2} \\ + -\frac{9}{2} \\ + \frac{11}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{13}{2},\frac{5}{2},\frac{13}{2},-2,10,\frac{3}{2},-\frac{9}{2},\frac{11}{2}\right)\, : \\ + \left\| \, \left(\frac{13}{2},\frac{5}{2},\frac{13}{2},-2,10,\frac{3}{2},-\frac{9}{2},\frac{11}{2}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{13}{2},\frac{5}{2},\frac{13}{2},-2,10,\frac{3}{2},-\frac{9}{2},\frac{11}{2}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{13}{2},\frac{5}{2},\frac{13}{2},-2,10,\frac{3}{2},-\frac{9}{2},\frac{11}{2}\right)\, \right\| =\sqrt{\left(\frac{13}{2}\right)^2+\left(\frac{5}{2}\right)^2+\left(\frac{13}{2}\right)^2+(-2)^2+10^2+\left(\frac{3}{2}\right)^2+\left(\frac{-9}{2}\right)^2+\left(\frac{11}{2}\right)^2}=3 \sqrt{\frac{55}{2}} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3664.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3664.txt new file mode 100644 index 0000000000000000000000000000000000000000..ebf1bdb7caeb07852c736f5f75fd9db40818da0f --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3664.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{15}{2} \\ + \frac{17}{4} \\ + \frac{33}{4} \\ + -6 \\ + -4 \\ + -7 \\ + \frac{17}{4} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{15}{2},\frac{17}{4},\frac{33}{4},-6,-4,-7,\frac{17}{4}\right)\, : \\ + \left\| \, \left(\frac{15}{2},\frac{17}{4},\frac{33}{4},-6,-4,-7,\frac{17}{4}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{15}{2},\frac{17}{4},\frac{33}{4},-6,-4,-7,\frac{17}{4}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{15}{2},\frac{17}{4},\frac{33}{4},-6,-4,-7,\frac{17}{4}\right)\, \right\| =\sqrt{\left(\frac{15}{2}\right)^2+\left(\frac{17}{4}\right)^2+\left(\frac{33}{4}\right)^2+(-6)^2+(-4)^2+(-7)^2+\left(\frac{17}{4}\right)^2}=\frac{\sqrt{4183}}{4} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3707.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3707.txt new file mode 100644 index 0000000000000000000000000000000000000000..3294f9bf4b6208cc073a36d9a3268a1f1eba2363 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3707.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{31}{16} \\ + -\frac{67}{16} \\ + -\frac{21}{4} \\ + -\frac{95}{16} \\ + -\frac{17}{4} \\ + -6 \\ + \frac{27}{16} \\ + \frac{93}{16} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{31}{16},-\frac{67}{16},-\frac{21}{4},-\frac{95}{16},-\frac{17}{4},-6,\frac{27}{16},\frac{93}{16}\right)\, : \\ + \left\| \, \left(-\frac{31}{16},-\frac{67}{16},-\frac{21}{4},-\frac{95}{16},-\frac{17}{4},-6,\frac{27}{16},\frac{93}{16}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{31}{16},-\frac{67}{16},-\frac{21}{4},-\frac{95}{16},-\frac{17}{4},-6,\frac{27}{16},\frac{93}{16}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{31}{16},-\frac{67}{16},-\frac{21}{4},-\frac{95}{16},-\frac{17}{4},-6,\frac{27}{16},\frac{93}{16}\right)\, \right\| =\sqrt{\left(\frac{-31}{16}\right)^2+\left(\frac{-67}{16}\right)^2+\left(\frac{-21}{4}\right)^2+\left(\frac{-95}{16}\right)^2+\left(\frac{-17}{4}\right)^2+(-6)^2+\left(\frac{27}{16}\right)^2+\left(\frac{93}{16}\right)^2}=\frac{\sqrt{44749}}{16} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3838.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3838.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b0a78620f8c38dc0a167cd409a9a1e8edb1e81a --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3838.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -5 \\ + 3 \\ + -\frac{13}{3} \\ + -\frac{2}{3} \\ + -\frac{20}{3} \\ + \frac{29}{3} \\ + \frac{4}{3} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-5,3,-\frac{13}{3},-\frac{2}{3},-\frac{20}{3},\frac{29}{3},\frac{4}{3}\right)\, : \\ + \left\| \, \left(-5,3,-\frac{13}{3},-\frac{2}{3},-\frac{20}{3},\frac{29}{3},\frac{4}{3}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-5,3,-\frac{13}{3},-\frac{2}{3},-\frac{20}{3},\frac{29}{3},\frac{4}{3}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-5,3,-\frac{13}{3},-\frac{2}{3},-\frac{20}{3},\frac{29}{3},\frac{4}{3}\right)\, \right\| =\sqrt{(-5)^2+3^2+\left(\frac{-13}{3}\right)^2+\left(\frac{-2}{3}\right)^2+\left(\frac{-20}{3}\right)^2+\left(\frac{29}{3}\right)^2+\left(\frac{4}{3}\right)^2}=\frac{2 \sqrt{434}}{3} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3880.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3880.txt new file mode 100644 index 0000000000000000000000000000000000000000..017875dc13bd0275b95fa436c4f7c3bb3bb1da31 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3880.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -3 \\ + -9 \\ + 1 \\ + 7 \\ + 1 \\ + 2 \\ + -2 \\ + 0 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-3,-9,1,7,1,2,-2,0)\, : \\ + \| \, (-3,-9,1,7,1,2,-2,0)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-3,-9,1,7,1,2,-2,0)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-3,-9,1,7,1,2,-2,0)\, \| =\sqrt{(-3)^2+(-9)^2+1^2+7^2+1^2+2^2+(-2)^2+0^2}=\sqrt{149} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3887.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3887.txt new file mode 100644 index 0000000000000000000000000000000000000000..7f7515620fbcf057b4e97a3499a83029fb0f0440 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/3887.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{63}{8} \\ + \frac{59}{8} \\ + \frac{13}{8} \\ + -\frac{11}{8} \\ + -\frac{5}{2} \\ + 7 \\ + \frac{21}{8} \\ + -\frac{35}{4} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{63}{8},\frac{59}{8},\frac{13}{8},-\frac{11}{8},-\frac{5}{2},7,\frac{21}{8},-\frac{35}{4}\right)\, : \\ + \left\| \, \left(\frac{63}{8},\frac{59}{8},\frac{13}{8},-\frac{11}{8},-\frac{5}{2},7,\frac{21}{8},-\frac{35}{4}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{63}{8},\frac{59}{8},\frac{13}{8},-\frac{11}{8},-\frac{5}{2},7,\frac{21}{8},-\frac{35}{4}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{63}{8},\frac{59}{8},\frac{13}{8},-\frac{11}{8},-\frac{5}{2},7,\frac{21}{8},-\frac{35}{4}\right)\, \right\| =\sqrt{\left(\frac{63}{8}\right)^2+\left(\frac{59}{8}\right)^2+\left(\frac{13}{8}\right)^2+\left(\frac{-11}{8}\right)^2+\left(\frac{-5}{2}\right)^2+7^2+\left(\frac{21}{8}\right)^2+\left(\frac{-35}{4}\right)^2}=\frac{\sqrt{16617}}{8} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4008.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4008.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bf04bb8a1e03ef59495d1e623a06692864e3370 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4008.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -2 \\ + 0 \\ + -8 \\ + 0 \\ + 6 \\ + 0 \\ + 8 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-2,0,-8,0,6,0,8)\, : \\ + \| \, (-2,0,-8,0,6,0,8)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-2,0,-8,0,6,0,8)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-2,0,-8,0,6,0,8)\, \| =\sqrt{(-2)^2+0^2+(-8)^2+0^2+6^2+0^2+8^2}=2 \sqrt{42} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4075.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4075.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b2a00b4417eea3a44240a306cc9844ce90d3fc3 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4075.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{11}{10} \\ + -\frac{7}{2} \\ + \frac{14}{5} \\ + \frac{61}{10} \\ + \frac{27}{5} \\ + -\frac{19}{2} \\ + \frac{67}{10} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{11}{10},-\frac{7}{2},\frac{14}{5},\frac{61}{10},\frac{27}{5},-\frac{19}{2},\frac{67}{10}\right)\, : \\ + \left\| \, \left(\frac{11}{10},-\frac{7}{2},\frac{14}{5},\frac{61}{10},\frac{27}{5},-\frac{19}{2},\frac{67}{10}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{11}{10},-\frac{7}{2},\frac{14}{5},\frac{61}{10},\frac{27}{5},-\frac{19}{2},\frac{67}{10}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{11}{10},-\frac{7}{2},\frac{14}{5},\frac{61}{10},\frac{27}{5},-\frac{19}{2},\frac{67}{10}\right)\, \right\| =\sqrt{\left(\frac{11}{10}\right)^2+\left(\frac{-7}{2}\right)^2+\left(\frac{14}{5}\right)^2+\left(\frac{61}{10}\right)^2+\left(\frac{27}{5}\right)^2+\left(\frac{-19}{2}\right)^2+\left(\frac{67}{10}\right)^2}=\frac{\sqrt{22281}}{10} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/414.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/414.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa9037837ea1ccf8e6ff7ce2cdfba0106837703c --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/414.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 6 \\ + 2 \\ + -1 \\ + -2 \\ + -5 \\ + -4 \\ + 6 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (6,2,-1,-2,-5,-4,6)\, : \\ + \| \, (6,2,-1,-2,-5,-4,6)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (6,2,-1,-2,-5,-4,6)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (6,2,-1,-2,-5,-4,6)\, \| =\sqrt{6^2+2^2+(-1)^2+(-2)^2+(-5)^2+(-4)^2+6^2}=\sqrt{122} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4237.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4237.txt new file mode 100644 index 0000000000000000000000000000000000000000..e925c1f6e18af5701030eae29cc9f03281cc0246 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4237.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{14}{3} \\ + -\frac{16}{3} \\ + \frac{20}{3} \\ + \frac{11}{3} \\ + -\frac{50}{9} \\ + -\frac{71}{9} \\ + -\frac{16}{3} \\ + 3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{14}{3},-\frac{16}{3},\frac{20}{3},\frac{11}{3},-\frac{50}{9},-\frac{71}{9},-\frac{16}{3},3\right)\, : \\ + \left\| \, \left(-\frac{14}{3},-\frac{16}{3},\frac{20}{3},\frac{11}{3},-\frac{50}{9},-\frac{71}{9},-\frac{16}{3},3\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{14}{3},-\frac{16}{3},\frac{20}{3},\frac{11}{3},-\frac{50}{9},-\frac{71}{9},-\frac{16}{3},3\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{14}{3},-\frac{16}{3},\frac{20}{3},\frac{11}{3},-\frac{50}{9},-\frac{71}{9},-\frac{16}{3},3\right)\, \right\| =\sqrt{\left(\frac{-14}{3}\right)^2+\left(\frac{-16}{3}\right)^2+\left(\frac{20}{3}\right)^2+\left(\frac{11}{3}\right)^2+\left(\frac{-50}{9}\right)^2+\left(\frac{-71}{9}\right)^2+\left(\frac{-16}{3}\right)^2+3^2}=\frac{\sqrt{19331}}{9} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4269.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4269.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3c9a747784713d766233d21b44f56e35d08562c --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4269.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 5 \\ + -3 \\ + 2 \\ + 2 \\ + 5 \\ + -1 \\ + -8 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (5,-3,2,2,5,-1,-8)\, : \\ + \| \, (5,-3,2,2,5,-1,-8)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (5,-3,2,2,5,-1,-8)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (5,-3,2,2,5,-1,-8)\, \| =\sqrt{5^2+(-3)^2+2^2+2^2+5^2+(-1)^2+(-8)^2}=2 \sqrt{33} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4304.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4304.txt new file mode 100644 index 0000000000000000000000000000000000000000..01f576e0e7c11441276a7a80e949885a76fe9579 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4304.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 5 \\ + 7 \\ + -2 \\ + -2 \\ + 5 \\ + -7 \\ + 1 \\ + -10 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (5,7,-2,-2,5,-7,1,-10)\, : \\ + \| \, (5,7,-2,-2,5,-7,1,-10)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (5,7,-2,-2,5,-7,1,-10)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (5,7,-2,-2,5,-7,1,-10)\, \| =\sqrt{5^2+7^2+(-2)^2+(-2)^2+5^2+(-7)^2+1^2+(-10)^2}=\sqrt{257} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4306.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4306.txt new file mode 100644 index 0000000000000000000000000000000000000000..11d5ed001ca7ab2a9495c01ac06c5e3dbe895aaa --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4306.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -4 \\ + -\frac{5}{4} \\ + -\frac{7}{4} \\ + 4 \\ + \frac{17}{4} \\ + -\frac{3}{4} \\ + \frac{29}{4} \\ + -\frac{11}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-4,-\frac{5}{4},-\frac{7}{4},4,\frac{17}{4},-\frac{3}{4},\frac{29}{4},-\frac{11}{2}\right)\, : \\ + \left\| \, \left(-4,-\frac{5}{4},-\frac{7}{4},4,\frac{17}{4},-\frac{3}{4},\frac{29}{4},-\frac{11}{2}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-4,-\frac{5}{4},-\frac{7}{4},4,\frac{17}{4},-\frac{3}{4},\frac{29}{4},-\frac{11}{2}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-4,-\frac{5}{4},-\frac{7}{4},4,\frac{17}{4},-\frac{3}{4},\frac{29}{4},-\frac{11}{2}\right)\, \right\| =\sqrt{(-4)^2+\left(\frac{-5}{4}\right)^2+\left(\frac{-7}{4}\right)^2+4^2+\left(\frac{17}{4}\right)^2+\left(\frac{-3}{4}\right)^2+\left(\frac{29}{4}\right)^2+\left(\frac{-11}{2}\right)^2}=\frac{47}{4} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4308.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4308.txt new file mode 100644 index 0000000000000000000000000000000000000000..8c273538b4c7b6a0bab57a5a139ed3a10e9de20a --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4308.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 1 \\ + -\frac{1}{4} \\ + 1 \\ + \frac{11}{2} \\ + \frac{39}{4} \\ + \frac{1}{4} \\ + \frac{19}{2} \\ + 1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(1,-\frac{1}{4},1,\frac{11}{2},\frac{39}{4},\frac{1}{4},\frac{19}{2},1\right)\, : \\ + \left\| \, \left(1,-\frac{1}{4},1,\frac{11}{2},\frac{39}{4},\frac{1}{4},\frac{19}{2},1\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(1,-\frac{1}{4},1,\frac{11}{2},\frac{39}{4},\frac{1}{4},\frac{19}{2},1\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(1,-\frac{1}{4},1,\frac{11}{2},\frac{39}{4},\frac{1}{4},\frac{19}{2},1\right)\, \right\| =\sqrt{1^2+\left(\frac{-1}{4}\right)^2+1^2+\left(\frac{11}{2}\right)^2+\left(\frac{39}{4}\right)^2+\left(\frac{1}{4}\right)^2+\left(\frac{19}{2}\right)^2+1^2}=\frac{\sqrt{3499}}{4} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4425.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4425.txt new file mode 100644 index 0000000000000000000000000000000000000000..cfe9df3e06ad4c1dd77e89295d97df43b10586e0 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4425.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -3 \\ + -6 \\ + -2 \\ + -1 \\ + 5 \\ + -3 \\ + -3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-3,-6,-2,-1,5,-3,-3)\, : \\ + \| \, (-3,-6,-2,-1,5,-3,-3)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-3,-6,-2,-1,5,-3,-3)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-3,-6,-2,-1,5,-3,-3)\, \| =\sqrt{(-3)^2+(-6)^2+(-2)^2+(-1)^2+5^2+(-3)^2+(-3)^2}=\sqrt{93} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/453.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/453.txt new file mode 100644 index 0000000000000000000000000000000000000000..59cc27c8a9b1af95dd3e5fa6c4cdb3b1f2eea8d0 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/453.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -5 \\ + 8 \\ + -6 \\ + -9 \\ + -4 \\ + -3 \\ + 2 \\ + -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-5,8,-6,-9,-4,-3,2,-2)\, : \\ + \| \, (-5,8,-6,-9,-4,-3,2,-2)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-5,8,-6,-9,-4,-3,2,-2)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-5,8,-6,-9,-4,-3,2,-2)\, \| =\sqrt{(-5)^2+8^2+(-6)^2+(-9)^2+(-4)^2+(-3)^2+2^2+(-2)^2}=\sqrt{239} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4562.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4562.txt new file mode 100644 index 0000000000000000000000000000000000000000..2349e2ce433dd8a3d0cb414bcf211caac572d3b9 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4562.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{1}{2} \\ + -3 \\ + -\frac{11}{2} \\ + 6 \\ + -\frac{13}{2} \\ + -6 \\ + -1 \\ + 6 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{1}{2},-3,-\frac{11}{2},6,-\frac{13}{2},-6,-1,6\right)\, : \\ + \left\| \, \left(\frac{1}{2},-3,-\frac{11}{2},6,-\frac{13}{2},-6,-1,6\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{1}{2},-3,-\frac{11}{2},6,-\frac{13}{2},-6,-1,6\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{1}{2},-3,-\frac{11}{2},6,-\frac{13}{2},-6,-1,6\right)\, \right\| =\sqrt{\left(\frac{1}{2}\right)^2+(-3)^2+\left(\frac{-11}{2}\right)^2+6^2+\left(\frac{-13}{2}\right)^2+(-6)^2+(-1)^2+6^2}=\frac{\sqrt{763}}{2} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4607.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4607.txt new file mode 100644 index 0000000000000000000000000000000000000000..1ed5fc8330eff267ff98479381d710be79620103 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4607.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{41}{8} \\ + \frac{73}{8} \\ + \frac{19}{2} \\ + \frac{1}{2} \\ + \frac{39}{8} \\ + -\frac{33}{8} \\ + -\frac{27}{4} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{41}{8},\frac{73}{8},\frac{19}{2},\frac{1}{2},\frac{39}{8},-\frac{33}{8},-\frac{27}{4}\right)\, : \\ + \left\| \, \left(\frac{41}{8},\frac{73}{8},\frac{19}{2},\frac{1}{2},\frac{39}{8},-\frac{33}{8},-\frac{27}{4}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{41}{8},\frac{73}{8},\frac{19}{2},\frac{1}{2},\frac{39}{8},-\frac{33}{8},-\frac{27}{4}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{41}{8},\frac{73}{8},\frac{19}{2},\frac{1}{2},\frac{39}{8},-\frac{33}{8},-\frac{27}{4}\right)\, \right\| =\sqrt{\left(\frac{41}{8}\right)^2+\left(\frac{73}{8}\right)^2+\left(\frac{19}{2}\right)^2+\left(\frac{1}{2}\right)^2+\left(\frac{39}{8}\right)^2+\left(\frac{-33}{8}\right)^2+\left(\frac{-27}{4}\right)^2}=\frac{\sqrt{\frac{2291}{2}}}{2} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4766.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4766.txt new file mode 100644 index 0000000000000000000000000000000000000000..2611cf046a8cbd6ff157d2e2fda26ef3ecd67210 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4766.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 10 \\ + 9 \\ + -3 \\ + 3 \\ + 8 \\ + -3 \\ + 2 \\ + -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (10,9,-3,3,8,-3,2,-2)\, : \\ + \| \, (10,9,-3,3,8,-3,2,-2)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (10,9,-3,3,8,-3,2,-2)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (10,9,-3,3,8,-3,2,-2)\, \| =\sqrt{10^2+9^2+(-3)^2+3^2+8^2+(-3)^2+2^2+(-2)^2}=2 \sqrt{70} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/480.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/480.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a94a27900c59ecf34fa3c0c3dbde72275d449b8 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/480.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 8 \\ + 9 \\ + 4 \\ + -8 \\ + 7 \\ + 4 \\ + -5 \\ + -9 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (8,9,4,-8,7,4,-5,-9)\, : \\ + \| \, (8,9,4,-8,7,4,-5,-9)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (8,9,4,-8,7,4,-5,-9)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (8,9,4,-8,7,4,-5,-9)\, \| =\sqrt{8^2+9^2+4^2+(-8)^2+7^2+4^2+(-5)^2+(-9)^2}=6 \sqrt{11} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4821.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4821.txt new file mode 100644 index 0000000000000000000000000000000000000000..0509534ebb572c4ebbf44a93fe9d9dc963d75789 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4821.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 2 \\ + 0 \\ + 7 \\ + 5 \\ + -1 \\ + 0 \\ + -1 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (2,0,7,5,-1,0,-1)\, : \\ + \| \, (2,0,7,5,-1,0,-1)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (2,0,7,5,-1,0,-1)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (2,0,7,5,-1,0,-1)\, \| =\sqrt{2^2+0^2+7^2+5^2+(-1)^2+0^2+(-1)^2}=4 \sqrt{5} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4842.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4842.txt new file mode 100644 index 0000000000000000000000000000000000000000..0dbaae239bb4a9e3ff89c8a109af8d4deea1a857 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/4842.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{22}{7} \\ + \frac{33}{7} \\ + -\frac{67}{7} \\ + -\frac{2}{7} \\ + -\frac{53}{7} \\ + \frac{68}{7} \\ + \frac{4}{7} \\ + -\frac{1}{7} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{22}{7},\frac{33}{7},-\frac{67}{7},-\frac{2}{7},-\frac{53}{7},\frac{68}{7},\frac{4}{7},-\frac{1}{7}\right)\, : \\ + \left\| \, \left(\frac{22}{7},\frac{33}{7},-\frac{67}{7},-\frac{2}{7},-\frac{53}{7},\frac{68}{7},\frac{4}{7},-\frac{1}{7}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{22}{7},\frac{33}{7},-\frac{67}{7},-\frac{2}{7},-\frac{53}{7},\frac{68}{7},\frac{4}{7},-\frac{1}{7}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{22}{7},\frac{33}{7},-\frac{67}{7},-\frac{2}{7},-\frac{53}{7},\frac{68}{7},\frac{4}{7},-\frac{1}{7}\right)\, \right\| =\sqrt{\left(\frac{22}{7}\right)^2+\left(\frac{33}{7}\right)^2+\left(\frac{-67}{7}\right)^2+\left(\frac{-2}{7}\right)^2+\left(\frac{-53}{7}\right)^2+\left(\frac{68}{7}\right)^2+\left(\frac{4}{7}\right)^2+\left(\frac{-1}{7}\right)^2}=\frac{2 \sqrt{3379}}{7} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/502.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/502.txt new file mode 100644 index 0000000000000000000000000000000000000000..0927ff91a4c4ee74add5e32d0794f4badd218bb9 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/502.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 9 \\ + 9 \\ + -9 \\ + 0 \\ + 8 \\ + 2 \\ + -4 \\ + 5 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (9,9,-9,0,8,2,-4,5)\, : \\ + \| \, (9,9,-9,0,8,2,-4,5)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (9,9,-9,0,8,2,-4,5)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (9,9,-9,0,8,2,-4,5)\, \| =\sqrt{9^2+9^2+(-9)^2+0^2+8^2+2^2+(-4)^2+5^2}=4 \sqrt{22} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/585.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/585.txt new file mode 100644 index 0000000000000000000000000000000000000000..078858920718d908d83907f40abfcc2a2ffb1500 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/585.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{29}{3} \\ + -\frac{10}{3} \\ + -\frac{14}{3} \\ + -\frac{7}{3} \\ + 9 \\ + \frac{26}{3} \\ + -\frac{25}{3} \\ + \frac{22}{3} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{29}{3},-\frac{10}{3},-\frac{14}{3},-\frac{7}{3},9,\frac{26}{3},-\frac{25}{3},\frac{22}{3}\right)\, : \\ + \left\| \, \left(\frac{29}{3},-\frac{10}{3},-\frac{14}{3},-\frac{7}{3},9,\frac{26}{3},-\frac{25}{3},\frac{22}{3}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{29}{3},-\frac{10}{3},-\frac{14}{3},-\frac{7}{3},9,\frac{26}{3},-\frac{25}{3},\frac{22}{3}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{29}{3},-\frac{10}{3},-\frac{14}{3},-\frac{7}{3},9,\frac{26}{3},-\frac{25}{3},\frac{22}{3}\right)\, \right\| =\sqrt{\left(\frac{29}{3}\right)^2+\left(\frac{-10}{3}\right)^2+\left(\frac{-14}{3}\right)^2+\left(\frac{-7}{3}\right)^2+9^2+\left(\frac{26}{3}\right)^2+\left(\frac{-25}{3}\right)^2+\left(\frac{22}{3}\right)^2}=\frac{10 \sqrt{37}}{3} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/611.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/611.txt new file mode 100644 index 0000000000000000000000000000000000000000..7ac4900a123d244ffc4e62580c281b72f70f998d --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/611.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{5}{4} \\ + -\frac{1}{2} \\ + -\frac{5}{4} \\ + \frac{63}{8} \\ + -\frac{1}{8} \\ + -\frac{3}{2} \\ + \frac{23}{4} \\ + -\frac{25}{8} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{5}{4},-\frac{1}{2},-\frac{5}{4},\frac{63}{8},-\frac{1}{8},-\frac{3}{2},\frac{23}{4},-\frac{25}{8}\right)\, : \\ + \left\| \, \left(-\frac{5}{4},-\frac{1}{2},-\frac{5}{4},\frac{63}{8},-\frac{1}{8},-\frac{3}{2},\frac{23}{4},-\frac{25}{8}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{5}{4},-\frac{1}{2},-\frac{5}{4},\frac{63}{8},-\frac{1}{8},-\frac{3}{2},\frac{23}{4},-\frac{25}{8}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{5}{4},-\frac{1}{2},-\frac{5}{4},\frac{63}{8},-\frac{1}{8},-\frac{3}{2},\frac{23}{4},-\frac{25}{8}\right)\, \right\| =\sqrt{\left(\frac{-5}{4}\right)^2+\left(\frac{-1}{2}\right)^2+\left(\frac{-5}{4}\right)^2+\left(\frac{63}{8}\right)^2+\left(\frac{-1}{8}\right)^2+\left(\frac{-3}{2}\right)^2+\left(\frac{23}{4}\right)^2+\left(\frac{-25}{8}\right)^2}=\frac{\sqrt{7071}}{8} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/630.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/630.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1eb3efade78d2707e64ac43b92370081e6d908e --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/630.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{11}{3} \\ + \frac{19}{3} \\ + -\frac{19}{3} \\ + \frac{28}{3} \\ + \frac{5}{3} \\ + \frac{25}{3} \\ + -5 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{11}{3},\frac{19}{3},-\frac{19}{3},\frac{28}{3},\frac{5}{3},\frac{25}{3},-5\right)\, : \\ + \left\| \, \left(-\frac{11}{3},\frac{19}{3},-\frac{19}{3},\frac{28}{3},\frac{5}{3},\frac{25}{3},-5\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{11}{3},\frac{19}{3},-\frac{19}{3},\frac{28}{3},\frac{5}{3},\frac{25}{3},-5\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{11}{3},\frac{19}{3},-\frac{19}{3},\frac{28}{3},\frac{5}{3},\frac{25}{3},-5\right)\, \right\| =\sqrt{\left(\frac{-11}{3}\right)^2+\left(\frac{19}{3}\right)^2+\left(\frac{-19}{3}\right)^2+\left(\frac{28}{3}\right)^2+\left(\frac{5}{3}\right)^2+\left(\frac{25}{3}\right)^2+(-5)^2}=\sqrt{278} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/640.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/640.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e85ac4d1e571aea19bcae3c42f0ea55630f777d --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/640.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 6 \\ + 8 \\ + -2 \\ + -10 \\ + -5 \\ + -8 \\ + -3 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (6,8,-2,-10,-5,-8,-3)\, : \\ + \| \, (6,8,-2,-10,-5,-8,-3)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (6,8,-2,-10,-5,-8,-3)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (6,8,-2,-10,-5,-8,-3)\, \| =\sqrt{6^2+8^2+(-2)^2+(-10)^2+(-5)^2+(-8)^2+(-3)^2}=\sqrt{302} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/732.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/732.txt new file mode 100644 index 0000000000000000000000000000000000000000..1603d8e7e8f42acb732ccc1c71151d348a069f47 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/732.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 6 \\ + -6 \\ + -1 \\ + 10 \\ + 8 \\ + -9 \\ + 8 \\ + -9 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (6,-6,-1,10,8,-9,8,-9)\, : \\ + \| \, (6,-6,-1,10,8,-9,8,-9)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (6,-6,-1,10,8,-9,8,-9)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (6,-6,-1,10,8,-9,8,-9)\, \| =\sqrt{6^2+(-6)^2+(-1)^2+10^2+8^2+(-9)^2+8^2+(-9)^2}=\sqrt{463} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/761.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/761.txt new file mode 100644 index 0000000000000000000000000000000000000000..020b20ef111dfb9d78a40a083c0ed2c81b5045fd --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/761.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{53}{7} \\ + \frac{66}{7} \\ + -\frac{68}{7} \\ + \frac{50}{7} \\ + -\frac{46}{7} \\ + -\frac{5}{7} \\ + \frac{55}{7} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{53}{7},\frac{66}{7},-\frac{68}{7},\frac{50}{7},-\frac{46}{7},-\frac{5}{7},\frac{55}{7}\right)\, : \\ + \left\| \, \left(-\frac{53}{7},\frac{66}{7},-\frac{68}{7},\frac{50}{7},-\frac{46}{7},-\frac{5}{7},\frac{55}{7}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{53}{7},\frac{66}{7},-\frac{68}{7},\frac{50}{7},-\frac{46}{7},-\frac{5}{7},\frac{55}{7}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{53}{7},\frac{66}{7},-\frac{68}{7},\frac{50}{7},-\frac{46}{7},-\frac{5}{7},\frac{55}{7}\right)\, \right\| =\sqrt{\left(\frac{-53}{7}\right)^2+\left(\frac{66}{7}\right)^2+\left(\frac{-68}{7}\right)^2+\left(\frac{50}{7}\right)^2+\left(\frac{-46}{7}\right)^2+\left(\frac{-5}{7}\right)^2+\left(\frac{55}{7}\right)^2}=\frac{\sqrt{19455}}{7} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/790.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/790.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad58e64b58d2a2a187546a7730551303e618983e --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/790.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + 9 \\ + 0 \\ + -7 \\ + -9 \\ + -2 \\ + 7 \\ + 6 \\ + -2 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (9,0,-7,-9,-2,7,6,-2)\, : \\ + \| \, (9,0,-7,-9,-2,7,6,-2)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (9,0,-7,-9,-2,7,6,-2)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (9,0,-7,-9,-2,7,6,-2)\, \| =\sqrt{9^2+0^2+(-7)^2+(-9)^2+(-2)^2+7^2+6^2+(-2)^2}=4 \sqrt{19} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/814.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/814.txt new file mode 100644 index 0000000000000000000000000000000000000000..50875f96be7be3cb51e1c6372ab4952556a45082 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/814.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -2 \\ + -6 \\ + -1 \\ + -8 \\ + -3 \\ + 6 \\ + 2 \\ + -7 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-2,-6,-1,-8,-3,6,2,-7)\, : \\ + \| \, (-2,-6,-1,-8,-3,6,2,-7)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-2,-6,-1,-8,-3,6,2,-7)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-2,-6,-1,-8,-3,6,2,-7)\, \| =\sqrt{(-2)^2+(-6)^2+(-1)^2+(-8)^2+(-3)^2+6^2+2^2+(-7)^2}=\sqrt{203} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/896.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/896.txt new file mode 100644 index 0000000000000000000000000000000000000000..b69800a1d666cee56ee9e3e7d22d5566d54a4aa1 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/896.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -1 \\ + 7 \\ + -1 \\ + 2 \\ + -3 \\ + 0 \\ + 5 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, (-1,7,-1,2,-3,0,5)\, : \\ + \| \, (-1,7,-1,2,-3,0,5)\, \| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, (-1,7,-1,2,-3,0,5)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \| \, (-1,7,-1,2,-3,0,5)\, \| =\sqrt{(-1)^2+7^2+(-1)^2+2^2+(-3)^2+0^2+5^2}=\sqrt{89} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/909.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/909.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3e1735722b5947dca4fc62bc9ea9b05e865fef7 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/909.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -\frac{11}{2} \\ + \frac{1}{8} \\ + \frac{9}{4} \\ + \frac{57}{8} \\ + -\frac{71}{8} \\ + \frac{45}{8} \\ + \frac{13}{4} \\ + -10 \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-\frac{11}{2},\frac{1}{8},\frac{9}{4},\frac{57}{8},-\frac{71}{8},\frac{45}{8},\frac{13}{4},-10\right)\, : \\ + \left\| \, \left(-\frac{11}{2},\frac{1}{8},\frac{9}{4},\frac{57}{8},-\frac{71}{8},\frac{45}{8},\frac{13}{4},-10\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-\frac{11}{2},\frac{1}{8},\frac{9}{4},\frac{57}{8},-\frac{71}{8},\frac{45}{8},\frac{13}{4},-10\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-\frac{11}{2},\frac{1}{8},\frac{9}{4},\frac{57}{8},-\frac{71}{8},\frac{45}{8},\frac{13}{4},-10\right)\, \right\| =\sqrt{\left(\frac{-11}{2}\right)^2+\left(\frac{1}{8}\right)^2+\left(\frac{9}{4}\right)^2+\left(\frac{57}{8}\right)^2+\left(\frac{-71}{8}\right)^2+\left(\frac{45}{8}\right)^2+\left(\frac{13}{4}\right)^2+(-10)^2}=\frac{17 \sqrt{17}}{4} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/910.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/910.txt new file mode 100644 index 0000000000000000000000000000000000000000..362d6de3ced19b3d36b47bde878436de30e1fbfc --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/910.txt @@ -0,0 +1,40 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + \frac{35}{4} \\ + \frac{17}{2} \\ + \frac{7}{2} \\ + -\frac{7}{4} \\ + -\frac{31}{4} \\ + \frac{7}{2} \\ + -\frac{13}{4} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(\frac{35}{4},\frac{17}{2},\frac{7}{2},-\frac{7}{4},-\frac{31}{4},\frac{7}{2},-\frac{13}{4}\right)\, : \\ + \left\| \, \left(\frac{35}{4},\frac{17}{2},\frac{7}{2},-\frac{7}{4},-\frac{31}{4},\frac{7}{2},-\frac{13}{4}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{7-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{7-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(\frac{35}{4},\frac{17}{2},\frac{7}{2},-\frac{7}{4},-\frac{31}{4},\frac{7}{2},-\frac{13}{4}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(\frac{35}{4},\frac{17}{2},\frac{7}{2},-\frac{7}{4},-\frac{31}{4},\frac{7}{2},-\frac{13}{4}\right)\, \right\| =\sqrt{\left(\frac{35}{4}\right)^2+\left(\frac{17}{2}\right)^2+\left(\frac{7}{2}\right)^2+\left(\frac{-7}{4}\right)^2+\left(\frac{-31}{4}\right)^2+\left(\frac{7}{2}\right)^2+\left(\frac{-13}{4}\right)^2}=\sqrt{247} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array} diff --git a/pretraining/mathematica/linear_algebra/lp_norm_w_steps/964.txt b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/964.txt new file mode 100644 index 0000000000000000000000000000000000000000..bdf03dc8fa2d3ff13eb7d997b8ce7234709dd778 --- /dev/null +++ b/pretraining/mathematica/linear_algebra/lp_norm_w_steps/964.txt @@ -0,0 +1,41 @@ +Problem: +Find the $\ell_2$ norm of the following vector: +$\left( +\begin{array}{c} + -3 \\ + -\frac{13}{2} \\ + \frac{5}{2} \\ + 7 \\ + 2 \\ + \frac{1}{2} \\ + 0 \\ + \frac{15}{2} \\ +\end{array} +\right)$. +Answer: +\begin{array}{l} + +\begin{array}{l} + \text{Find the norm of the vector }\, \left(-3,-\frac{13}{2},\frac{5}{2},7,2,\frac{1}{2},0,\frac{15}{2}\right)\, : \\ + \left\| \, \left(-3,-\frac{13}{2},\frac{5}{2},7,2,\frac{1}{2},0,\frac{15}{2}\right)\, \right\| \\ +\end{array} + \\ +\hline + +\begin{array}{l} + \text{The }\text{formula }\text{for }\text{the }\text{8-dimensional }\text{Euclidean }\text{norm }\text{comes }\text{from }\text{the }\text{8-dimensional }\text{Pythagorean }\text{theorem}: \\ + \left\| \, \left(v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\right)\, \right\| =\sqrt{v_1^2+v_2^2+v_3^2+v_4^2+v_5^2+v_6^2+v_7^2+v_8^2} \\ +\end{array} + \\ + +\begin{array}{l} + \text{Substitute }\, \left(-3,-\frac{13}{2},\frac{5}{2},7,2,\frac{1}{2},0,\frac{15}{2}\right)\, \text{into }\text{the }\text{formula}: \\ + \fbox{$ +\begin{array}{ll} + \text{Answer:} & \\ + \text{} & \left\| \, \left(-3,-\frac{13}{2},\frac{5}{2},7,2,\frac{1}{2},0,\frac{15}{2}\right)\, \right\| =\sqrt{(-3)^2+\left(\frac{-13}{2}\right)^2+\left(\frac{5}{2}\right)^2+7^2+2^2+\left(\frac{1}{2}\right)^2+0^2+\left(\frac{15}{2}\right)^2}=\sqrt{167} \\ +\end{array} +$} \\ +\end{array} + \\ +\end{array}