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pretraining/mathematica/linear_algebra/null_space/10081.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ 3 & 9 & 3 & -10 & -6 \\
+ -9 & -6 & 10 & 4 & -8 \\
+ -2 & -4 & 1 & 8 & -1 \\
+ -5 & -3 & -8 & 3 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${\{-8632.,8379.,-1045.,2589.,3415.\}}$

pretraining/mathematica/linear_algebra/null_space/10449.txt

@@ -0,0 +1,9 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ 0 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${\{1.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/11032.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ -10 & -7 \\
+ -1 & 4 \\
+ 1 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/11656.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ -1 & -6 \\
+ -4 & -8 \\
+ 3 & 6 \\
+ -7 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/12305.txt

@@ -0,0 +1,13 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ -5 & 2 & -8 & 6 \\
+ -9 & -2 & -9 & 10 \\
+ 3 & -4 & -5 & 4 \\
+ -4 & 0 & -2 & 2 \\
+ 0 & 6 & -10 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/12408.txt

@@ -0,0 +1,10 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 8 & 0 & 7 \\
+ 8 & 9 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${\{-63.,136.,72.\}}$

pretraining/mathematica/linear_algebra/null_space/13106.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ -2 & -9 \\
+ 2 & -3 \\
+ 10 & 8 \\
+ -3 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/13159.txt

@@ -0,0 +1,9 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 0 & 4 & 7 \\
+\end{array}
+\right)$.
+Answer:
+${\{0.,-7.,4.\}, \{1.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/13745.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ 5 & -6 & -8 & 1 \\
+ -2 & 9 & 3 & 10 \\
+ 1 & 1 & -2 & -3 \\
+ -8 & -7 & -2 & 4 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/15833.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ -10 & -8 & -2 & 7 \\
+ -9 & 7 & -10 & -2 \\
+ -3 & 5 & 9 & -9 \\
+ -9 & 2 & 6 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/16019.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ -7 & -2 & -10 & -8 & -9 \\
+ -5 & -9 & 9 & 10 & 9 \\
+ 9 & 1 & 7 & -9 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${\{1332.,-1353.,-1101.,0.,488.\}, \{1588.,-1535.,-1195.,488.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/16426.txt

@@ -0,0 +1,13 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ 2 & -8 \\
+ 5 & -10 \\
+ -1 & -7 \\
+ 8 & -2 \\
+ 0 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/16540.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ -5 & 7 \\
+ 0 & -4 \\
+ -7 & -8 \\
+ 6 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/16807.txt

@@ -0,0 +1,10 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ -9 & 4 & -7 & 2 \\
+ 3 & 9 & 8 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-95.,-51.,93.,0.\}, \{-6.,-29.,0.,31.\}}$

pretraining/mathematica/linear_algebra/null_space/17349.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ -10 & -4 & 7 \\
+ 1 & -10 & 0 \\
+ -3 & 9 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/17723.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ 7 & -6 & -5 & 5 & 8 \\
+ 9 & 8 & -7 & 5 & 10 \\
+ -10 & -2 & -3 & 0 & 4 \\
+ 8 & 3 & 1 & -5 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${\{385.,-75.,3140.,-2737.,3280.\}}$

pretraining/mathematica/linear_algebra/null_space/18130.txt

@@ -0,0 +1,10 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ -5 & -10 & -3 \\
+ 0 & 7 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${\{69.,-45.,35.\}}$

pretraining/mathematica/linear_algebra/null_space/18443.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ 10 & -7 & 6 & 5 \\
+ 8 & 8 & 0 & 2 \\
+ 3 & 10 & -10 & -6 \\
+ -9 & -8 & 9 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/20002.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ -1 & 10 & 7 \\
+ 1 & 2 & 9 \\
+ 8 & 1 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/2040.txt

@@ -0,0 +1,10 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ -3 & -5 & 0 \\
+ -10 & 7 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${\{30.,-18.,71.\}}$

pretraining/mathematica/linear_algebra/null_space/20501.txt

@@ -0,0 +1,9 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 7 & -8 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-9.,0.,7.\}, \{8.,7.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/22168.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ -5 & -4 & -1 & -5 \\
+ -3 & -5 & 3 & 0 \\
+ 3 & 6 & 2 & 3 \\
+ 10 & 9 & -7 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/23477.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ -2 & -7 & -7 & 7 \\
+ -5 & 10 & 5 & -2 \\
+ -9 & -1 & 10 & -7 \\
+\end{array}
+\right)$.
+Answer:
+${\{168.,-57.,464.,455.\}}$

pretraining/mathematica/linear_algebra/null_space/2361.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 4 & 9 & -6 \\
+ -3 & 5 & -1 \\
+ -3 & -9 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/24019.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ -8 & 7 & -1 & -8 & 2 \\
+ -4 & -8 & -9 & 10 & -6 \\
+ -9 & 6 & 1 & 5 & 3 \\
+ 4 & 4 & 0 & 8 & 4 \\
+\end{array}
+\right)$.
+Answer:
+${\{-429.,-1664.,-454.,-298.,2689.\}}$

pretraining/mathematica/linear_algebra/null_space/24238.txt

@@ -0,0 +1,10 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ 5 & -1 & -2 & -1 & 2 \\
+ -10 & -5 & -3 & -3 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${\{-13.,5.,0.,0.,35.\}, \{1.,-5.,5.,0.,0.\}, \{2.,-25.,0.,35.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/24658.txt

@@ -0,0 +1,13 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ 2 & 10 \\
+ 4 & -10 \\
+ -1 & 8 \\
+ -1 & -1 \\
+ -5 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/24769.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 3 & 5 & -3 \\
+ -1 & -2 & -4 \\
+ 6 & 3 & 3 \\
+ -8 & -7 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/25070.txt

@@ -0,0 +1,13 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ 7 & 4 & -9 & -6 \\
+ 9 & 6 & -6 & -1 \\
+ 0 & -9 & 3 & -2 \\
+ -2 & -1 & 2 & 1 \\
+ -9 & -1 & -9 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/25434.txt

@@ -0,0 +1,10 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 9 & -3 & 9 \\
+ -6 & 3 & -7 \\
+\end{array}
+\right)$.
+Answer:
+${\{-2.,3.,3.\}}$

pretraining/mathematica/linear_algebra/null_space/2563.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ 6 & 6 & 3 & -5 \\
+ -1 & 9 & 2 & -10 \\
+ -6 & -3 & 5 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${\{0.,4.,-3.,3.\}}$

pretraining/mathematica/linear_algebra/null_space/2734.txt

@@ -0,0 +1,10 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ 9 & -6 & -10 & 9 & -6 \\
+ 7 & 9 & -8 & 8 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${\{-43.,-3.,0.,41.,0.\}, \{22.,-8.,0.,0.,41.\}, \{138.,2.,123.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/29209.txt

@@ -0,0 +1,10 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ -10 & -8 & 8 & -3 & 3 \\
+ 10 & 1 & -3 & 9 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-69.,60.,0.,70.,0.\}, \{8.,25.,35.,0.,0.\}, \{69.,-60.,0.,0.,70.\}}$

pretraining/mathematica/linear_algebra/null_space/29397.txt

@@ -0,0 +1,9 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 9 & -3 & 10 \\
+\end{array}
+\right)$.
+Answer:
+${\{-10.,0.,9.\}, \{1.,3.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/3032.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ -6 & -3 & 2 & 5 \\
+ -1 & -4 & 2 & -9 \\
+ 4 & 9 & -4 & 7 \\
+ -1 & 10 & 2 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/31286.txt

@@ -0,0 +1,10 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ 1 & -5 & 5 & -6 & -1 \\
+ -2 & -8 & 3 & 7 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-25.,13.,18.,0.,0.\}, \{-11.,-4.,0.,0.,9.\}, \{83.,-5.,0.,18.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/3133.txt

@@ -0,0 +1,13 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ 9 & -4 & -4 & -4 \\
+ -3 & -5 & -10 & -4 \\
+ 5 & 7 & 9 & 7 \\
+ -2 & -9 & 9 & 3 \\
+ -1 & 9 & 5 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/31387.txt

@@ -0,0 +1,13 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 8 & 2 & -6 \\
+ 10 & 1 & -10 \\
+ -6 & 1 & 0 \\
+ 10 & 7 & 1 \\
+ 3 & -7 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/31695.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 9 & -9 & -4 \\
+ -6 & 4 & -2 \\
+ -2 & -5 & 1 \\
+ 8 & -3 & -5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/32771.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ -8 & 6 & -6 & -3 \\
+ -10 & 4 & 6 & 9 \\
+ -10 & 8 & 0 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${\{-48.,-126.,-95.,66.\}}$

pretraining/mathematica/linear_algebra/null_space/33093.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ 10 & -5 & 9 & 0 & 8 \\
+ 3 & 5 & -6 & -6 & 9 \\
+ -8 & -2 & -2 & 5 & -1 \\
+ 3 & 0 & -5 & 5 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${\{-378.,4575.,2351.,2303.,687.\}}$

pretraining/mathematica/linear_algebra/null_space/33478.txt

@@ -0,0 +1,13 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ -6 & -9 \\
+ -9 & -2 \\
+ -1 & 5 \\
+ -7 & 5 \\
+ -8 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/33772.txt

@@ -0,0 +1,12 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccccc}
+ 9 & -8 & -3 & -9 & 6 \\
+ -3 & -5 & -5 & 3 & -1 \\
+ -10 & 0 & 6 & -8 & 3 \\
+ 9 & -7 & -2 & 8 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1366.,3612.,-2689.,4025.,11558.\}}$

pretraining/mathematica/linear_algebra/null_space/33851.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ -2 & 9 & 8 & 6 \\
+ 4 & -7 & 2 & 10 \\
+ 2 & -10 & -2 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-35.,-6.,-11.,12.\}}$

pretraining/mathematica/linear_algebra/null_space/34148.txt

@@ -0,0 +1,13 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cccc}
+ 7 & -1 & 3 & -9 \\
+ 3 & 4 & -8 & 7 \\
+ 3 & -5 & 10 & 10 \\
+ 0 & -3 & -10 & 4 \\
+ 0 & -1 & 3 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/34423.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ -6 & 2 & 0 \\
+ 10 & 7 & 7 \\
+ 1 & -2 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/34685.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ 9 & -5 \\
+ 4 & 1 \\
+ -1 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/36036.txt

@@ -0,0 +1,13 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ -8 & 3 \\
+ -7 & -7 \\
+ 6 & -3 \\
+ -5 & -1 \\
+ -1 & -4 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/36880.txt

@@ -0,0 +1,11 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{ccc}
+ 4 & -3 & -10 \\
+ 1 & 3 & 1 \\
+ -3 & 1 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/38438.txt

@@ -0,0 +1,9 @@
+Problem:
+Give a list of vectors that forms a basis for the null space of the following matrix (round your answer to three decimal places):
+$\left(
+\begin{array}{cc}
+ -8 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${\{-5.,4.\}}$