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pretraining/mathematica/linear_algebra/null_space/10293.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):
+$\left(
+\begin{array}{ccccc}
+ -9 & -8 & -9 & -3 & -6 \\
+ 1 & -6 & -4 & -5 & -1 \\
+ 5 & -9 & 3 & -4 & 8 \\
+ -4 & 7 & -8 & 3 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1448.,1191.,665.,-2409.,791.\}}$

pretraining/mathematica/linear_algebra/null_space/10965.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):
+$\left(
+\begin{array}{ccc}
+ 6 & 7 & -9 \\
+ 1 & 9 & -4 \\
+ 1 & -3 & 8 \\
+ 6 & -6 & -5 \\
+ -8 & -6 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/11335.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):
+$\left(
+\begin{array}{ccc}
+ -8 & 2 & -3 \\
+ 4 & -8 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-3.,-6.,4.\}}$

pretraining/mathematica/linear_algebra/null_space/11830.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):
+$\left(
+\begin{array}{cc}
+ -4 & 6 \\
+ 2 & -10 \\
+ -1 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/13112.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):
+$\left(
+\begin{array}{cccc}
+ 0 & -4 & 10 & 2 \\
+ -8 & 3 & -10 & -7 \\
+\end{array}
+\right)$.
+Answer:
+${\{-11.,8.,0.,16.\}, \{-5.,40.,16.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/15665.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):
+$\left(
+\begin{array}{cccc}
+ -7 & 10 & 1 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${\{-3.,0.,0.,7.\}, \{1.,0.,7.,0.\}, \{10.,7.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/19099.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):
+$\left(
+\begin{array}{cccc}
+ -3 & 9 & 10 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${\{1.,0.,0.,3.\}, \{3.,1.,0.,0.\}, \{10.,0.,3.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/19309.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):
+$\left(
+\begin{array}{cccc}
+ -3 & 6 & 7 & 10 \\
+ -5 & 7 & -1 & 8 \\
+ 1 & 9 & 9 & 3 \\
+\end{array}
+\right)$.
+Answer:
+${\{627.,54.,-229.,316.\}}$

pretraining/mathematica/linear_algebra/null_space/19816.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):
+$\left(
+\begin{array}{cc}
+ -1 & -9 \\
+ 6 & 6 \\
+ 3 & 1 \\
+ 9 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/21816.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):
+$\left(
+\begin{array}{cccc}
+ 5 & -5 & 9 & 0 \\
+ -3 & -8 & 9 & 4 \\
+\end{array}
+\right)$.
+Answer:
+${\{-27.,72.,55.,0.\}, \{4.,4.,0.,11.\}}$

pretraining/mathematica/linear_algebra/null_space/2595.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):
+$\left(
+\begin{array}{cc}
+ -2 & -7 \\
+ -9 & -2 \\
+ 8 & -7 \\
+ 5 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/27363.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):
+$\left(
+\begin{array}{cc}
+ 2 & -7 \\
+ 5 & 8 \\
+ -9 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29763.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):
+$\left(
+\begin{array}{ccccc}
+ 1 & -3 & 8 & 10 & 10 \\
+ -1 & 5 & 9 & -5 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-67.,-17.,2.,0.,0.\}, \{-35.,-5.,0.,2.,0.\}, \{-16.,-2.,0.,0.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/30462.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):
+$\left(
+\begin{array}{cccc}
+ -2 & -1 & -7 & -9 \\
+ -3 & -5 & 10 & 10 \\
+ -10 & 3 & 3 & 4 \\
+ 3 & 9 & -3 & 7 \\
+ -4 & -9 & 6 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/31810.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):
+$\left(
+\begin{array}{cc}
+ 6 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-3.,2.\}}$

pretraining/mathematica/linear_algebra/null_space/33299.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):
+$\left(
+\begin{array}{ccccc}
+ -9 & 7 & -8 & -1 & 5 \\
+ 6 & -10 & 9 & -6 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${\{-17.,33.,48.,0.,0.\}, \{-13.,-15.,0.,12.,0.\}, \{-1.,-7.,0.,0.,8.\}}$

pretraining/mathematica/linear_algebra/null_space/34556.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):
+$\left(
+\begin{array}{cc}
+ 3 & 10 \\
+ 9 & -5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/36353.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):
+$\left(
+\begin{array}{ccc}
+ 5 & -3 & 1 \\
+ -1 & -6 & 6 \\
+ 9 & -9 & 9 \\
+ 9 & 10 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/36507.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):
+$\left(
+\begin{array}{ccc}
+ 3 & -2 & -10 \\
+ -1 & -5 & 10 \\
+ 8 & 1 & -4 \\
+ -9 & -5 & 3 \\
+ -5 & 7 & -7 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/37186.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):
+$\left(
+\begin{array}{ccc}
+ 1 & 7 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-7.,1.,0.\}, \{1.,0.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/37626.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):
+$\left(
+\begin{array}{cccc}
+ -5 & 3 & 8 & -2 \\
+ -9 & 6 & 9 & 3 \\
+\end{array}
+\right)$.
+Answer:
+${\{-7.,-11.,0.,1.\}, \{7.,9.,1.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/38351.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):
+$\left(
+\begin{array}{cccc}
+ -6 & 6 & -1 & 0 \\
+ 5 & -3 & -10 & 10 \\
+ 2 & -5 & -2 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${\{334.,380.,276.,223.\}}$

pretraining/mathematica/linear_algebra/null_space/40602.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):
+$\left(
+\begin{array}{cccc}
+ -9 & 7 & -8 & -5 \\
+ 1 & -6 & -7 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-97.,-71.,47.,0.\}, \{12.,49.,0.,47.\}}$

pretraining/mathematica/linear_algebra/null_space/4120.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):
+$\left(
+\begin{array}{cccc}
+ -1 & 0 & 1 & -2 \\
+ -2 & -8 & 0 & 4 \\
+ -2 & -2 & 4 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-64.,21.,-44.,10.\}}$

pretraining/mathematica/linear_algebra/null_space/43259.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):
+$\left(
+\begin{array}{ccccc}
+ -4 & 9 & -10 & -2 & -8 \\
+ -6 & 0 & 5 & 4 & 8 \\
+ 3 & 2 & -7 & -4 & -7 \\
+ 3 & 0 & -6 & 2 & -8 \\
+ -7 & -3 & 10 & -10 & -4 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/4419.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):
+$\left(
+\begin{array}{cc}
+ 2 & -1 \\
+ -1 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/44371.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):
+$\left(
+\begin{array}{ccccc}
+ 1 & 2 & 10 & 1 & 10 \\
+ -9 & 2 & 7 & 2 & -3 \\
+ -9 & -3 & -3 & 8 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-12.,56.,-25.,0.,15.\}, \{81.,692.,-175.,285.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/44931.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):
+$\left(
+\begin{array}{cccc}
+ -9 & 0 & -1 & -7 \\
+ -8 & 0 & -9 & 3 \\
+ 10 & 1 & -8 & 7 \\
+ 3 & 2 & 5 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/4506.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):
+$\left(
+\begin{array}{ccccc}
+ 10 & 10 & 1 & 2 & 6 \\
+ -1 & 5 & -5 & 0 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-55.,49.,60.,0.,0.\}, \{-5.,-4.,0.,0.,15.\}, \{-5.,-1.,0.,30.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/45750.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):
+$\left(
+\begin{array}{cccc}
+ -1 & -6 & 9 & -6 \\
+ -10 & -1 & -1 & 10 \\
+\end{array}
+\right)$.
+Answer:
+${\{-15.,91.,59.,0.\}, \{66.,-70.,0.,59.\}}$

pretraining/mathematica/linear_algebra/null_space/47783.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):
+$\left(
+\begin{array}{ccc}
+ -9 & -1 & 6 \\
+ -2 & 5 & 8 \\
+ 8 & 6 & -8 \\
+ 3 & -6 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/48727.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):
+$\left(
+\begin{array}{ccc}
+ -4 & -1 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,0.,4.\}, \{-1.,4.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/5333.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):
+$\left(
+\begin{array}{cc}
+ -3 & 0 \\
+ 5 & 4 \\
+ 1 & -4 \\
+ 1 & -10 \\
+ -1 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/6715.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):
+$\left(
+\begin{array}{ccccc}
+ -8 & 7 & 7 & 0 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-9.,0.,0.,0.,8.\}, \{0.,0.,0.,1.,0.\}, \{7.,0.,8.,0.,0.\}, \{7.,8.,0.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/6896.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):
+$\left(
+\begin{array}{ccc}
+ 8 & 1 & -3 \\
+ -1 & 8 & 7 \\
+ 9 & -5 & 3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/7267.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):
+$\left(
+\begin{array}{cccc}
+ -8 & 5 & 2 & 7 \\
+ -3 & 0 & 10 & 10 \\
+ -5 & 0 & -7 & 0 \\
+ 2 & -7 & 1 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/7548.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):
+$\left(
+\begin{array}{ccccc}
+ 2 & 9 & -7 & 7 & -4 \\
+ -5 & -8 & 7 & -5 & 2 \\
+ 0 & -5 & -8 & 3 & -9 \\
+ 5 & -8 & 8 & 1 & -3 \\
+ 0 & -4 & -7 & 8 & 10 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/8937.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):
+$\left(
+\begin{array}{ccccc}
+ -5 & -5 & 9 & 8 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-6.,0.,0.,0.,5.\}, \{-1.,1.,0.,0.,0.\}, \{8.,0.,0.,5.,0.\}, \{9.,0.,5.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/961.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):
+$\left(
+\begin{array}{cccc}
+ 8 & 8 & -10 & -6 \\
+ -7 & 9 & 9 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${\{35.,13.,0.,64.\}, \{81.,-1.,64.,0.\}}$

pretraining/mathematica/number_theory/mod/11389.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 52416 \pmod{42}$.
+Answer:
+$0$

pretraining/mathematica/number_theory/mod/12736.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 18087 \pmod{94}$.
+Answer:
+$39$

pretraining/mathematica/number_theory/mod/14095.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 35671 \pmod{30}$.
+Answer:
+$1$

pretraining/mathematica/number_theory/mod/16437.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 91518 \pmod{7}$.
+Answer:
+$0$

pretraining/mathematica/number_theory/mod/17941.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 24092 \pmod{92}$.
+Answer:
+$80$

pretraining/mathematica/number_theory/mod/21770.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 81867 \pmod{51}$.
+Answer:
+$12$

pretraining/mathematica/number_theory/mod/22530.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 77000 \pmod{4}$.
+Answer:
+$0$

pretraining/mathematica/number_theory/mod/22742.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 10534 \pmod{72}$.
+Answer:
+$22$

pretraining/mathematica/number_theory/mod/23182.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 92947 \pmod{63}$.
+Answer:
+$22$

pretraining/mathematica/number_theory/mod/23954.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 10064 \pmod{82}$.
+Answer:
+$60$

pretraining/mathematica/number_theory/mod/24903.txt

@@ -0,0 +1,4 @@
+Problem:
+Find the smallest $x$ such that $x \equiv 24063 \pmod{88}$.
+Answer:
+$39$