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pretraining/mathematica/linear_algebra/null_space/1001.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):
+$\left(
+\begin{array}{ccc}
+ -8 & -8 & -3 \\
+ 7 & -8 & 9 \\
+ -10 & 0 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/10077.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):
+$\left(
+\begin{array}{cccc}
+ -10 & 5 & -1 & 5 \\
+ 6 & -6 & -1 & -7 \\
+ -8 & -7 & -5 & -3 \\
+ -7 & 2 & -7 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/10108.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):
+$\left(
+\begin{array}{cccc}
+ 4 & -2 & -7 & 7 \\
+ 9 & 9 & -1 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${\{-79.,31.,0.,54.\}, \{65.,-59.,54.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/11259.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):
+$\left(
+\begin{array}{ccccc}
+ -10 & 1 & 10 & 10 & 4 \\
+ 10 & 5 & 3 & -6 & 7 \\
+\end{array}
+\right)$.
+Answer:
+${\{13.,-110.,0.,0.,60.\}, \{14.,-10.,0.,15.,0.\}, \{47.,-130.,60.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/119.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):
+$\left(
+\begin{array}{ccccc}
+ 6 & -6 & -9 & 6 & -7 \\
+ 8 & -8 & -7 & 1 & -5 \\
+ 7 & -4 & -9 & -7 & 8 \\
+ 9 & 0 & 0 & 2 & 4 \\
+\end{array}
+\right)$.
+Answer:
+${\{-2740.,-6307.,1554.,3762.,4284.\}}$

pretraining/mathematica/linear_algebra/null_space/11951.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):
+$\left(
+\begin{array}{cccc}
+ -9 & 3 & -10 & -6 \\
+ 3 & 6 & 8 & -10 \\
+ -10 & 6 & 5 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${\{18.,1600.,-78.,903.\}}$

pretraining/mathematica/linear_algebra/null_space/12691.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):
+$\left(
+\begin{array}{ccccc}
+ 6 & -3 & -5 & 1 & -2 \\
+ 2 & -9 & -8 & 1 & 4 \\
+ -2 & -9 & 1 & -2 & -4 \\
+ -1 & -6 & -9 & -8 & 3 \\
+\end{array}
+\right)$.
+Answer:
+${\{4305.,-1436.,4086.,-2754.,3477.\}}$

pretraining/mathematica/linear_algebra/null_space/12857.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):
+$\left(
+\begin{array}{ccccc}
+ 3 & 5 & -7 & 10 & 5 \\
+ -6 & -9 & 2 & -7 & 1 \\
+ -6 & 8 & -3 & 4 & 7 \\
+ 5 & -8 & -5 & -8 & 7 \\
+ 0 & -3 & -7 & 0 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/14136.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):
+$\left(
+\begin{array}{cccc}
+ 0 & -10 & 0 & 2 \\
+ -4 & -4 & 7 & 9 \\
+ -8 & 5 & 7 & 7 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,4.,-24.,20.\}}$

pretraining/mathematica/linear_algebra/null_space/14283.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):
+$\left(
+\begin{array}{ccccc}
+ -2 & 1 & -1 & 10 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,0.,2.,0.,0.\}, \{1.,0.,0.,0.,2.\}, \{1.,2.,0.,0.,0.\}, \{5.,0.,0.,1.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/15957.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):
+$\left(
+\begin{array}{cc}
+ -8 & 1 \\
+ 10 & 9 \\
+ -3 & 7 \\
+ 4 & -3 \\
+ 0 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/1611.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):
+$\left(
+\begin{array}{ccc}
+ -6 & 9 & 5 \\
+ 2 & -9 & 1 \\
+ -5 & -4 & -5 \\
+ -9 & -6 & -5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/16633.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):
+$\left(
+\begin{array}{ccc}
+ -8 & -7 & -7 \\
+ 7 & 7 & 7 \\
+ -3 & -9 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/17327.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):
+$\left(
+\begin{array}{ccccc}
+ -2 & 3 & -10 & 10 & 6 \\
+ 8 & 1 & -3 & 0 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${\{-9.,-32.,0.,0.,13.\}, \{-1.,86.,26.,0.,0.\}, \{5.,-40.,0.,13.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/17675.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):
+$\left(
+\begin{array}{ccc}
+ 7 & 9 & -7 \\
+\end{array}
+\right)$.
+Answer:
+${\{-9.,7.,0.\}, \{1.,0.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/18365.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):
+$\left(
+\begin{array}{ccc}
+ -6 & -9 & 1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-3.,2.,0.\}, \{1.,0.,6.\}}$

pretraining/mathematica/linear_algebra/null_space/18690.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):
+$\left(
+\begin{array}{ccc}
+ 10 & -2 & -3 \\
+ -7 & -7 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${\{5.,-101.,84.\}}$

pretraining/mathematica/linear_algebra/null_space/19124.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):
+$\left(
+\begin{array}{ccc}
+ -7 & 1 & -5 \\
+ -1 & 6 & -6 \\
+ -7 & 5 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/19482.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):
+$\left(
+\begin{array}{cccc}
+ 4 & -8 & 5 & 1 \\
+ -5 & 0 & -2 & -3 \\
+ 9 & -2 & -4 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${\{-154.,-111.,-122.,338.\}}$

pretraining/mathematica/linear_algebra/null_space/19689.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):
+$\left(
+\begin{array}{ccc}
+ -6 & -4 & 2 \\
+ -5 & -1 & 4 \\
+ 0 & 1 & 10 \\
+ -2 & 8 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/20533.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):
+$\left(
+\begin{array}{ccc}
+ -4 & -9 & -9 \\
+ -6 & -9 & 8 \\
+ 8 & 8 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/20662.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):
+$\left(
+\begin{array}{ccc}
+ -2 & -4 & -9 \\
+ 4 & 5 & -2 \\
+ 7 & 5 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/20773.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):
+$\left(
+\begin{array}{ccc}
+ -2 & 10 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${\{0.,0.,1.\}, \{5.,1.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/21694.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):
+$\left(
+\begin{array}{ccc}
+ -7 & -5 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${\{-5.,7.,0.\}, \{0.,0.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/21855.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):
+$\left(
+\begin{array}{ccc}
+ 2 & -9 & -10 \\
+ -4 & 3 & 3 \\
+\end{array}
+\right)$.
+Answer:
+${\{-3.,-34.,30.\}}$

pretraining/mathematica/linear_algebra/null_space/22377.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):
+$\left(
+\begin{array}{ccc}
+ 4 & 0 & -8 \\
+ -2 & -10 & -5 \\
+\end{array}
+\right)$.
+Answer:
+${\{20.,-9.,10.\}}$

pretraining/mathematica/linear_algebra/null_space/24914.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):
+$\left(
+\begin{array}{cc}
+ 10 & 3 \\
+ 3 & -7 \\
+ -2 & 0 \\
+ 5 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/25101.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):
+$\left(
+\begin{array}{cc}
+ 10 & 0 \\
+ 4 & -9 \\
+ -1 & -4 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/25116.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):
+$\left(
+\begin{array}{ccccc}
+ -2 & -2 & 7 & -4 & 4 \\
+ 0 & 10 & -10 & 7 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${\{-13.,-7.,0.,10.,0.\}, \{5.,2.,2.,0.,0.\}, \{9.,1.,0.,0.,5.\}}$

pretraining/mathematica/linear_algebra/null_space/25176.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):
+$\left(
+\begin{array}{cc}
+ -6 & -5 \\
+ -7 & -3 \\
+ -8 & 0 \\
+ 6 & 3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/26373.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):
+$\left(
+\begin{array}{ccc}
+ 1 & -7 & 7 \\
+\end{array}
+\right)$.
+Answer:
+${\{-7.,0.,1.\}, \{7.,1.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/26706.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):
+$\left(
+\begin{array}{ccccc}
+ 1 & -3 & 5 & 5 & 6 \\
+ 5 & -8 & 2 & -2 & -2 \\
+ 7 & 1 & -3 & -8 & 6 \\
+ -2 & -6 & -3 & -6 & 1 \\
+ 0 & 2 & 3 & -9 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/26819.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):
+$\left(
+\begin{array}{cccc}
+ 8 & 0 & 8 & 5 \\
+ 10 & -2 & -1 & 5 \\
+ -7 & 9 & 7 & 1 \\
+ -10 & 5 & -2 & 4 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/28965.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):
+$\left(
+\begin{array}{cc}
+ -8 & 1 \\
+ -7 & 3 \\
+ -7 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29264.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):
+$\left(
+\begin{array}{cccc}
+ 9 & -1 & 9 & -7 \\
+ -8 & -3 & -10 & 10 \\
+ -9 & 2 & 8 & 1 \\
+ -2 & 7 & -10 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29539.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):
+$\left(
+\begin{array}{ccc}
+ 6 & -3 & 10 \\
+ 0 & -6 & 8 \\
+ 9 & -4 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29684.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):
+$\left(
+\begin{array}{ccc}
+ -10 & 7 & 9 \\
+ 3 & 6 & 4 \\
+ 1 & -6 & -7 \\
+ -4 & 8 & -4 \\
+ 7 & 6 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29785.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):
+$\left(
+\begin{array}{cccc}
+ -1 & -4 & 3 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-4.,1.,0.,0.\}, \{3.,0.,1.,0.\}, \{9.,0.,0.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/30959.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):
+$\left(
+\begin{array}{ccccc}
+ -9 & 4 & 7 & 8 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,0.,0.,0.,3.\}, \{4.,9.,0.,0.,0.\}, \{7.,0.,9.,0.,0.\}, \{8.,0.,0.,9.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/32981.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):
+$\left(
+\begin{array}{cc}
+ 6 & 8 \\
+ -3 & 7 \\
+ -3 & 2 \\
+ -5 & -5 \\
+ 3 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/33513.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):
+$\left(
+\begin{array}{ccc}
+ -4 & -3 & -3 \\
+ 7 & 2 & 6 \\
+ 0 & -9 & -2 \\
+ 4 & 0 & -2 \\
+ -4 & -3 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/33582.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):
+$\left(
+\begin{array}{cccc}
+ -10 & 0 & -6 & -9 \\
+ -4 & 0 & 7 & 9 \\
+ 5 & 9 & 7 & 8 \\
+ -9 & -10 & 7 & 0 \\
+ -3 & 1 & 5 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/33804.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):
+$\left(
+\begin{array}{ccccc}
+ 8 & 9 & -8 & -4 & 5 \\
+ -7 & 10 & -9 & -1 & -8 \\
+ 3 & -10 & 2 & -1 & 0 \\
+ 0 & 5 & -5 & 1 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-2172.,-911.,-1476.,-358.,2467.\}}$

pretraining/mathematica/linear_algebra/null_space/34595.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):
+$\left(
+\begin{array}{ccc}
+ -6 & -5 & 8 \\
+ -4 & 0 & 2 \\
+ -8 & 5 & -7 \\
+ -5 & 7 & 4 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/34680.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):
+$\left(
+\begin{array}{ccc}
+ 4 & 10 & 1 \\
+ -10 & 8 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-3.,-1.,22.\}}$

pretraining/mathematica/linear_algebra/null_space/35162.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):
+$\left(
+\begin{array}{ccc}
+ -9 & 0 & 10 \\
+ 4 & 4 & 3 \\
+ 0 & 0 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/36053.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):
+$\left(
+\begin{array}{cc}
+ -6 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,2.\}}$

pretraining/mathematica/linear_algebra/null_space/36491.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):
+$\left(
+\begin{array}{ccc}
+ 4 & -10 & 4 \\
+ 7 & -6 & -7 \\
+\end{array}
+\right)$.
+Answer:
+${\{47.,28.,23.\}}$

pretraining/mathematica/linear_algebra/null_space/377.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):
+$\left(
+\begin{array}{ccccc}
+ -2 & -9 & -4 & 6 & -3 \\
+ -5 & -3 & -3 & 10 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{-15.,-14.,39.,0.,0.\}, \{0.,-1.,0.,0.,3.\}, \{72.,10.,0.,39.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/37891.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):
+$\left(
+\begin{array}{cccc}
+ 9 & -8 & 5 & 3 \\
+ -7 & 5 & -3 & 0 \\
+ 6 & 9 & 5 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${\{22.,5.,-43.,19.\}}$