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pretraining/mathematica/linear_algebra/null_space/10187.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):
+$\left(
+\begin{array}{ccc}
+ -1 & -5 & 3 \\
+\end{array}
+\right)$.
+Answer:
+${\{-5.,1.,0.\}, \{3.,0.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/10246.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):
+$\left(
+\begin{array}{cccc}
+ 2 & 1 & 3 & 10 \\
+\end{array}
+\right)$.
+Answer:
+${\{-5.,0.,0.,1.\}, \{-3.,0.,2.,0.\}, \{-1.,2.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/10908.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):
+$\left(
+\begin{array}{ccc}
+ 2 & 1 & -8 \\
+ 3 & 0 & 4 \\
+\end{array}
+\right)$.
+Answer:
+${\{-4.,32.,3.\}}$

pretraining/mathematica/linear_algebra/null_space/11268.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):
+$\left(
+\begin{array}{cccc}
+ -6 & -10 & 2 & -4 \\
+ 10 & 2 & -5 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${\{-21.,-5.,0.,44.\}, \{23.,-5.,44.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/11943.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):
+$\left(
+\begin{array}{ccc}
+ 8 & 10 & 5 \\
+ -8 & -2 & 5 \\
+ -10 & 3 & 8 \\
+ 8 & 8 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/12273.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):
+$\left(
+\begin{array}{ccccc}
+ -3 & 0 & 4 & 6 & 5 \\
+ 10 & -7 & 7 & -3 & 10 \\
+ 8 & -10 & -2 & 3 & -4 \\
+ 1 & -4 & 10 & 7 & 7 \\
+\end{array}
+\right)$.
+Answer:
+${\{-2213.,-2251.,149.,-1320.,137.\}}$

pretraining/mathematica/linear_algebra/null_space/14102.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):
+$\left(
+\begin{array}{cccc}
+ 1 & 2 & -6 & 8 \\
+ -6 & 7 & -9 & 8 \\
+ -4 & -3 & -2 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-368.,-421.,157.,269.\}}$

pretraining/mathematica/linear_algebra/null_space/1482.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):
+$\left(
+\begin{array}{cc}
+ -3 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${\{-8.,3.\}}$

pretraining/mathematica/linear_algebra/null_space/15099.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):
+$\left(
+\begin{array}{ccccc}
+ 9 & 5 & 9 & 5 & 0 \\
+ -2 & 3 & 1 & 8 & -9 \\
+ -5 & 1 & 6 & 5 & -1 \\
+ 4 & -2 & 3 & -3 & 7 \\
+ 1 & -3 & -8 & -1 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/15717.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):
+$\left(
+\begin{array}{cccc}
+ 0 & -5 & 5 & 0 \\
+ 2 & 2 & 0 & 8 \\
+ 1 & -1 & 6 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-7.,-1.,-1.,2.\}}$

pretraining/mathematica/linear_algebra/null_space/159.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):
+$\left(
+\begin{array}{cc}
+ -7 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${\{-9.,7.\}}$

pretraining/mathematica/linear_algebra/null_space/16180.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):
+$\left(
+\begin{array}{ccc}
+ 8 & -3 & 1 \\
+ 8 & -4 & 6 \\
+ 9 & -10 & -1 \\
+ 3 & 2 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/16977.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):
+$\left(
+\begin{array}{cc}
+ 8 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${\{3.,8.\}}$

pretraining/mathematica/linear_algebra/null_space/17296.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):
+$\left(
+\begin{array}{cccc}
+ -1 & 1 & 2 & -2 \\
+ 3 & -6 & -6 & 1 \\
+ 8 & -1 & -6 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${\{68.,-50.,89.,30.\}}$

pretraining/mathematica/linear_algebra/null_space/17501.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):
+$\left(
+\begin{array}{cc}
+ 6 & 2 \\
+ 1 & -10 \\
+ -7 & 2 \\
+ 2 & 9 \\
+ -9 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/17602.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):
+$\left(
+\begin{array}{ccc}
+ -3 & 2 & 0 \\
+ -2 & 9 & -5 \\
+\end{array}
+\right)$.
+Answer:
+${\{10.,15.,23.\}}$

pretraining/mathematica/linear_algebra/null_space/18714.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):
+$\left(
+\begin{array}{cc}
+ -2 & 2 \\
+ 9 & -5 \\
+ 0 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/19283.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):
+$\left(
+\begin{array}{ccc}
+ -10 & 6 & -1 \\
+ 1 & -1 & 6 \\
+ -1 & -10 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/20554.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):
+$\left(
+\begin{array}{cccc}
+ 4 & 8 & 1 & 4 \\
+ 7 & 2 & -10 & -5 \\
+\end{array}
+\right)$.
+Answer:
+${\{1.,-1.,0.,1.\}, \{82.,-47.,48.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/21396.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):
+$\left(
+\begin{array}{cccc}
+ -6 & 8 & 3 & 8 \\
+\end{array}
+\right)$.
+Answer:
+${\{1.,0.,2.,0.\}, \{4.,0.,0.,3.\}, \{4.,3.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/21746.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):
+$\left(
+\begin{array}{ccc}
+ 8 & -1 & -4 \\
+ 0 & -9 & -5 \\
+ -5 & 0 & -2 \\
+ 0 & -2 & 5 \\
+ -9 & 0 & 7 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/22201.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):
+$\left(
+\begin{array}{ccc}
+ -8 & -2 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,4.,0.\}, \{0.,0.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/22330.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):
+$\left(
+\begin{array}{cccc}
+ -3 & 4 & -6 & -10 \\
+ 8 & 9 & 1 & -3 \\
+ -6 & 10 & -5 & -2 \\
+ -4 & -4 & -1 & -10 \\
+ 0 & -1 & 7 & 3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/22475.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):
+$\left(
+\begin{array}{ccccc}
+ -10 & -5 & 2 & -1 & 9 \\
+ 0 & 3 & -5 & 6 & 7 \\
+ -6 & 3 & -5 & -10 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-152.,421.,321.,57.,0.\}, \{-19.,404.,402.,0.,114.\}}$

pretraining/mathematica/linear_algebra/null_space/23853.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):
+$\left(
+\begin{array}{ccc}
+ 10 & 10 & -10 \\
+ 3 & 10 & -3 \\
+ 2 & 6 & 3 \\
+ 8 & -9 & -6 \\
+ 5 & -6 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/24506.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):
+$\left(
+\begin{array}{cc}
+ 9 & 2 \\
+ 10 & -3 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/24619.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):
+$\left(
+\begin{array}{ccc}
+ -9 & 6 & -5 \\
+ 6 & -9 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-3.,8.,15.\}}$

pretraining/mathematica/linear_algebra/null_space/24745.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):
+$\left(
+\begin{array}{ccc}
+ 5 & 10 & 4 \\
+ -8 & -3 & 1 \\
+ 9 & 4 & -3 \\
+ -5 & 0 & -2 \\
+ 4 & -9 & -9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/24953.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):
+$\left(
+\begin{array}{ccc}
+ -2 & -9 & 7 \\
+ 0 & 6 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-1.,1.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/25443.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):
+$\left(
+\begin{array}{cccc}
+ -1 & -5 & 10 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${\{-10.,0.,0.,1.\}, \{-5.,1.,0.,0.\}, \{10.,0.,1.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/25638.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):
+$\left(
+\begin{array}{ccccc}
+ 3 & 2 & 0 & 6 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${\{-2.,0.,0.,1.,0.\}, \{-2.,3.,0.,0.,0.\}, \{0.,0.,1.,0.,0.\}, \{2.,0.,0.,0.,3.\}}$

pretraining/mathematica/linear_algebra/null_space/26045.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):
+$\left(
+\begin{array}{ccc}
+ -9 & 4 & 8 \\
+ 7 & 6 & -1 \\
+ -6 & -9 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/26526.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):
+$\left(
+\begin{array}{cccc}
+ 4 & 0 & -6 & 5 \\
+ -4 & -5 & -6 & 1 \\
+ 3 & -5 & 10 & -3 \\
+ -10 & -1 & 3 & -7 \\
+ 6 & 8 & 7 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/26557.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):
+$\left(
+\begin{array}{cccc}
+ -8 & 9 & -9 & -10 \\
+ -5 & 1 & 8 & -5 \\
+ -10 & 4 & -3 & -1 \\
+ 3 & -6 & 3 & 7 \\
+ 6 & -7 & -4 & 5 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/2761.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):
+$\left(
+\begin{array}{ccccc}
+ 8 & -8 & -3 & -8 & -1 \\
+\end{array}
+\right)$.
+Answer:
+${\{1.,0.,0.,0.,8.\}, \{1.,0.,0.,1.,0.\}, \{1.,1.,0.,0.,0.\}, \{3.,0.,8.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/28044.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):
+$\left(
+\begin{array}{cc}
+ 1 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-6.,1.\}}$

pretraining/mathematica/linear_algebra/null_space/28456.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):
+$\left(
+\begin{array}{cccc}
+ -6 & 5 & -1 & -3 \\
+ 10 & 6 & 3 & -1 \\
+ 8 & 1 & 3 & -9 \\
+ -4 & -9 & 3 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29422.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):
+$\left(
+\begin{array}{ccccc}
+ -7 & 7 & -6 & 7 & -4 \\
+ -7 & 1 & 1 & -5 & 5 \\
+ -7 & 10 & 5 & -1 & -9 \\
+ -7 & -1 & 8 & -10 & -4 \\
+ 4 & -6 & 5 & -3 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/29572.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):
+$\left(
+\begin{array}{cc}
+ 7 & -4 \\
+ 7 & -8 \\
+ -9 & -9 \\
+ 7 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/30477.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):
+$\left(
+\begin{array}{cccc}
+ 0 & 3 & -7 & -9 \\
+ -1 & -9 & -2 & 3 \\
+ 8 & 3 & 10 & -4 \\
+ -5 & -3 & 1 & -7 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/30925.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):
+$\left(
+\begin{array}{ccc}
+ 4 & 8 & 2 \\
+ -6 & 4 & 8 \\
+ -8 & 3 & -7 \\
+ 6 & 0 & 6 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/31350.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):
+$\left(
+\begin{array}{cccc}
+ -5 & -5 & 5 & 4 \\
+ 1 & -9 & 7 & 4 \\
+ -7 & -1 & -9 & -8 \\
+\end{array}
+\right)$.
+Answer:
+${\{9.,-20.,-67.,70.\}}$

pretraining/mathematica/linear_algebra/null_space/32287.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):
+$\left(
+\begin{array}{cc}
+ 8 & 8 \\
+ 4 & -3 \\
+ -3 & 2 \\
+ -2 & -6 \\
+ -3 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/33434.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):
+$\left(
+\begin{array}{cc}
+ -1 & -3 \\
+ -10 & -1 \\
+ 7 & 4 \\
+ 4 & -4 \\
+ 2 & 2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/34048.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):
+$\left(
+\begin{array}{ccccc}
+ 8 & 0 & -6 & -4 & 10 \\
+ 4 & 5 & -6 & 5 & -10 \\
+\end{array}
+\right)$.
+Answer:
+${\{-5.,12.,0.,0.,4.\}, \{5.,-14.,0.,10.,0.\}, \{15.,12.,20.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/3421.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):
+$\left(
+\begin{array}{ccc}
+ 0 & 2 & -10 \\
+ 9 & -3 & 6 \\
+ -9 & 6 & -5 \\
+ 6 & 4 & -7 \\
+ 6 & 3 & -7 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/3488.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):
+$\left(
+\begin{array}{cccc}
+ -3 & 10 & -8 & 0 \\
+\end{array}
+\right)$.
+Answer:
+${\{-8.,0.,3.,0.\}, \{0.,0.,0.,1.\}, \{10.,3.,0.,0.\}}$

pretraining/mathematica/linear_algebra/null_space/35260.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):
+$\left(
+\begin{array}{cc}
+ -8 & 10 \\
+ 2 & 1 \\
+ -7 & 3 \\
+ -3 & 9 \\
+ -8 & -2 \\
+\end{array}
+\right)$.
+Answer:
+${}$

pretraining/mathematica/linear_algebra/null_space/35674.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):
+$\left(
+\begin{array}{cccc}
+ -3 & 9 & -9 & 7 \\
+ -1 & 10 & -8 & -6 \\
+\end{array}
+\right)$.
+Answer:
+${\{-6.,5.,7.,0.\}, \{124.,25.,0.,21.\}}$

pretraining/mathematica/linear_algebra/null_space/3647.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):
+$\left(
+\begin{array}{cccc}
+ -7 & -9 & 1 & -1 \\
+ 7 & 6 & -8 & -2 \\
+ 6 & 1 & -3 & 3 \\
+ 1 & -1 & -6 & -10 \\
+ 9 & 5 & 6 & 9 \\
+\end{array}
+\right)$.
+Answer:
+${}$