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pretraining/mathematica/linear_algebra/null_space/10781.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):
+$\left(
+\begin{array}{ccc}
+ 6 & 9 & -6 \\
+ 4 & -8 & 1 \\
+ -6 & 0 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/10977.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):
+$\left(
+\begin{array}{cc}
+ 7 & -2 \\
+ -10 & -4 \\
+ -3 & 0 \\
+ -10 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/13236.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):
+$\left(
+\begin{array}{cc}
+ 6 & 5 \\
+ 9 & 6 \\
+ 9 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/13443.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):
+$\left(
+\begin{array}{ccccc}
+ -9 & -8 & -7 & 4 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${\{-10.,0.,0.,0.,9.\}, \{-8.,9.,0.,0.,0.\}, \{-7.,0.,9.,0.,0.\}, \{4.,0.,0.,9.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/13518.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):
+$\left(
+\begin{array}{ccc}
+ -3 & 6 & -6 \\
+ 5 & 6 & 3 \\
+ -6 & 7 & -9 \\
+ 7 & -5 & -4 \\
+ 1 & -4 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/13838.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):
+$\left(
+\begin{array}{ccc}
+ 5 & -10 & 6 \\
+ -3 & -5 & 4 \\
+ 10 & -8 & 10 \\
+ 0 & -8 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/14346.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):
+$\left(
+\begin{array}{cc}
+ -5 & 3 \\
+ -3 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/15609.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):
+$\left(
+\begin{array}{ccc}
+ -8 & 2 & -3 \\
+ -2 & 0 & -7 \\
+ 3 & 4 & -5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/16190.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):
+$\left(
+\begin{array}{cccc}
+ 3 & 8 & -6 & -10 \\
+ 5 & -5 & -1 & -9 \\
+ 10 & -10 & 1 & 4 \\
+ 5 & -9 & 10 & 4 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/16305.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):
+$\left(
+\begin{array}{cc}
+ -5 & 10 \\
+ 4 & 6 \\
+ -8 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/17038.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):
+$\left(
+\begin{array}{ccc}
+ 0 & 6 & -6 \\
+ 9 & 9 & 1 \\
+ 2 & -9 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/17220.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):
+$\left(
+\begin{array}{ccccc}
+ -4 & 4 & -6 & -2 & 6 \\
+ 10 & 3 & -7 & 2 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-7.,6.,0.,26.,0.\}, \{5.,44.,26.,0.,0.\}, \{7.,-32.,0.,0.,26.\}}$

pretraining/mathematica/linear_algebra/null_space/17235.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):
+$\left(
+\begin{array}{ccc}
+ 1 & -10 & 10 \\
+ -4 & -6 & -5 \\
+ -9 & 10 & 8 \\
+ -3 & -1 & 7 \\
+ 5 & -5 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/18056.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):
+$\left(
+\begin{array}{cccc}
+ -3 & -2 & -10 & 10 \\
+ -5 & 9 & -3 & 0 \\
+ 10 & 7 & -1 & 6 \\
+ -8 & 8 & -8 & 3 \\
+ -2 & 0 & 0 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/19132.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):
+$\left(
+\begin{array}{cccc}
+ 4 & 2 & -6 & -4 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,2.,0.,0.\}, \{1.,0.,0.,1.\}, \{3.,0.,2.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/19350.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):
+$\left(
+\begin{array}{ccc}
+ 0 & -5 & 4 \\
+ 5 & 1 & -4 \\
+ 1 & 2 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/20333.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):
+$\left(
+\begin{array}{cccc}
+ 1 & 8 & -4 & 4 \\
+ 3 & 4 & 4 & -9 \\
+ 8 & -3 & -10 & 2 \\
+ 6 & 3 & -4 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/20615.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):
+$\left(
+\begin{array}{ccc}
+ 0 & -7 & 8 \\
+ 4 & -1 & -5 \\
+ -10 & 8 & 4 \\
+ -2 & 7 & -8 \\
+ -4 & -3 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/20785.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):
+$\left(
+\begin{array}{cccc}
+ -5 & -5 & 0 & -6 \\
+ 5 & 6 & -10 & 6 \\
+ 4 & 2 & -4 & 5 \\
+ -2 & 8 & 5 & -2 \\
+ 9 & 0 & -2 & -4 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/21092.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):
+$\left(
+\begin{array}{ccccc}
+ 0 & 10 & -5 & -9 & 1 \\
+ -8 & 8 & 1 & -6 & 10 \\
+ 8 & 5 & 8 & -4 & 4 \\
+ 7 & 8 & -6 & -2 & -2 \\
+ -8 & 10 & 4 & 3 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/22048.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):
+$\left(
+\begin{array}{cccc}
+ 8 & 4 & -4 & -5 \\
+ 8 & -1 & 6 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,4.,2.,0.\}, \{1.,8.,0.,8.\}}$

pretraining/mathematica/linear_algebra/null_space/22418.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):
+$\left(
+\begin{array}{cc}
+ -1 & -5 \\
+ 9 & -7 \\
+ 6 & 8 \\
+ -2 & -10 \\
+ 8 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/23425.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):
+$\left(
+\begin{array}{cc}
+ 5 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${\{2.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/23452.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):
+$\left(
+\begin{array}{ccccc}
+ 8 & -4 & 7 & 3 & -2 \\
+ 0 & -4 & 8 & -4 & 6 \\
+ -5 & 0 & 9 & 4 & 1 \\
+ -8 & 0 & -8 & -7 & -6 \\
+ -2 & -10 & -10 & 4 & 3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/23833.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):
+$\left(
+\begin{array}{ccc}
+ 10 & 8 & -6 \\
+ 1 & 0 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${\{18.,-21.,2.\}}$

pretraining/mathematica/linear_algebra/null_space/25379.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):
+$\left(
+\begin{array}{ccccc}
+ -4 & 10 & -1 & -10 & 10 \\
+ -2 & -1 & -10 & -6 & 2 \\
+ -9 & 1 & 2 & -3 & -9 \\
+ 4 & -10 & -5 & -8 & -9 \\
+ -9 & -2 & -1 & -1 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/25890.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):
+$\left(
+\begin{array}{cc}
+ 3 & -9 \\
+ 1 & 9 \\
+ 9 & 7 \\
+ -4 & 10 \\
+ -7 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/25962.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):
+$\left(
+\begin{array}{cccc}
+ 5 & 8 & -4 & -10 \\
+ -3 & 9 & -2 & -10 \\
+ 0 & 6 & -7 & -3 \\
+ 1 & -4 & 0 & -9 \\
+ -6 & -2 & 8 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/2605.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):
+$\left(
+\begin{array}{ccc}
+ -3 & -7 & -1 \\
+ -3 & 5 & -3 \\
+ -5 & -1 & 4 \\
+ 5 & 8 & -4 \\
+ -10 & 8 & -7 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/26216.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):
+$\left(
+\begin{array}{ccc}
+ 8 & 10 & -6 \\
+ 4 & 9 & -4 \\
+ 6 & -7 & 8 \\
+ 0 & -6 & 9 \\
+ -3 & 0 & 10 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/27669.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):
+$\left(
+\begin{array}{ccccc}
+ 2 & 4 & 10 & 9 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-9.,0.,0.,2.,0.\}, \{-5.,0.,1.,0.,0.\}, \{-2.,1.,0.,0.,0.\}, \{9.,0.,0.,0.,2.\}}$

pretraining/mathematica/linear_algebra/null_space/27800.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):
+$\left(
+\begin{array}{cccc}
+ 2 & 7 & 3 & -2 \\
+ 0 & 7 & 9 & -6 \\
+ 1 & -8 & 5 & 4 \\
+ -2 & 1 & -1 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29634.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):
+$\left(
+\begin{array}{ccc}
+ -6 & -4 & -9 \\
+ 2 & -8 & -7 \\
+ -3 & -9 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29766.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):
+$\left(
+\begin{array}{cc}
+ 8 & 5 \\
+ 6 & 3 \\
+ -5 & -8 \\
+ -7 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29898.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):
+$\left(
+\begin{array}{cc}
+ 10 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{1.,10.\}}$

pretraining/mathematica/linear_algebra/null_space/30481.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):
+$\left(
+\begin{array}{cccc}
+ 6 & -6 & 6 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,0.,1.,0.\}, \{1.,0.,0.,6.\}, \{1.,1.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/30725.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):
+$\left(
+\begin{array}{cccc}
+ 4 & -8 & 2 & 4 \\
+ 9 & 4 & 10 & -5 \\
+\end{array}
+\right)$.
+Answer:
+${\{-4.,-1.,4.,0.\}, \{3.,7.,0.,11.\}}$

pretraining/mathematica/linear_algebra/null_space/30979.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):
+$\left(
+\begin{array}{cccc}
+ -6 & -2 & 6 & 4 \\
+ 3 & -5 & -4 & -9 \\
+ -10 & 9 & 0 & 10 \\
+ 2 & -4 & 6 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/31042.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):
+$\left(
+\begin{array}{ccccc}
+ -7 & -9 & 10 & 8 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-9.,7.,0.,0.,0.\}, \{-1.,0.,0.,0.,7.\}, \{8.,0.,0.,7.,0.\}, \{10.,0.,7.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/31966.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):
+$\left(
+\begin{array}{ccc}
+ 0 & -5 & 2 \\
+ 7 & 9 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-13.,14.,35.\}}$

pretraining/mathematica/linear_algebra/null_space/3244.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):
+$\left(
+\begin{array}{ccc}
+ 5 & 5 & 7 \\
+ 2 & -5 & -7 \\
+ 1 & 2 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/33745.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):
+$\left(
+\begin{array}{ccccc}
+ 5 & 1 & -2 & -9 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${\{-8.,0.,0.,0.,5.\}, \{-1.,5.,0.,0.,0.\}, \{2.,0.,5.,0.,0.\}, \{9.,0.,0.,5.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/36531.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):
+$\left(
+\begin{array}{cccc}
+ -1 & -8 & 1 & 4 \\
+ 1 & 6 & -2 & 10 \\
+ -8 & 6 & -10 & 6 \\
+ -7 & 8 & 7 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/36634.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):
+$\left(
+\begin{array}{cccc}
+ -2 & -2 & 7 & -8 \\
+ -8 & -3 & -1 & 2 \\
+ -1 & -8 & 3 & -7 \\
+ 5 & 10 & 0 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/36966.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):
+$\left(
+\begin{array}{ccccc}
+ -2 & 6 & -3 & -2 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${\{-3.,0.,2.,0.,0.\}, \{-1.,0.,0.,1.,0.\}, \{1.,0.,0.,0.,1.\}, \{3.,1.,0.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/3860.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):
+$\left(
+\begin{array}{cccc}
+ 2 & -5 & -10 & 5 \\
+ 3 & 8 & 0 & 4 \\
+ -3 & 3 & -2 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-20.,-17.,29.,49.\}}$

pretraining/mathematica/linear_algebra/null_space/38725.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):
+$\left(
+\begin{array}{cccc}
+ -6 & 3 & 0 & -1 \\
+ 6 & -6 & -2 & 10 \\
+ 2 & -7 & 3 & 2 \\
+ 7 & 2 & 2 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/39584.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):
+$\left(
+\begin{array}{ccc}
+ 1 & -3 & 3 \\
+ 0 & 0 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${\{3.,1.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/4048.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):
+$\left(
+\begin{array}{cccc}
+ 10 & 3 & -4 & -8 \\
+ -9 & 1 & -9 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${\{-23.,126.,37.,0.\}, \{2.,92.,0.,37.\}}$

pretraining/mathematica/linear_algebra/null_space/40858.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):
+$\left(
+\begin{array}{cccc}
+ 2 & -6 & 10 & -5 \\
+ -6 & 6 & -1 & -4 \\
+\end{array}
+\right)$.
+Answer:
+${\{-27.,-19.,0.,12.\}, \{27.,29.,12.,0.\}}$