Difference between revisions of "031 Review Problems"

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−
'''1.''' True or false: If all the entries of a <math>7\times 7</math> matrix <math>A</math> are <math>7,</math> then det <math>A</math> must be <math>7^7.</math>
+
'''1.''' True or false: If all the entries of a &nbsp;<math style="vertical-align: 0px">7\times 7</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">7,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">\text{det }A</math>&nbsp; must be &nbsp;<math style="vertical-align: 0px">7^7.</math>
  
 
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−
'''2.''' True or false: If a matrix <math>A^2</math> is diagonalizable, then the matrix <math>A</math> must be diagonalizable as well.
+
'''2.''' True or false: If a matrix &nbsp;<math style="vertical-align: 0px">A^2</math>&nbsp; is diagonalizable, then the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; must be diagonalizable as well.
  
 
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|}
  
−
'''3.''' True or false: If <math>A</math> is a <math>4\times 4</math> matrix with characteristic equation <math>\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math> then <math>A</math> is diagonalizable.
+
'''3.''' True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a &nbsp;<math style="vertical-align: -1px">4\times 4</math>&nbsp; matrix with characteristic equation &nbsp;<math style="vertical-align: -5px">\lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is diagonalizable.
  
 
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|}
  
−
'''4.''' True or false: If <math>A</math> is invertible, then <math>A</math> is diagonalizable.
+
'''4.''' True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is invertible, then &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is diagonalizable.
  
 
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−
'''5.''' True or false: If <math>A</math> and <math>B</math> are invertible <math>n\times n</math> matrices, then so is <math>A+B.</math>
+
'''5.''' True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; are invertible &nbsp;<math style="vertical-align: 0px">n\times n</math>&nbsp; matrices, then so is &nbsp;<math style="vertical-align: -1px">A+B.</math>
  
 
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−
'''6.''' True or false: If <math>A</math> is a <math>3\times 5</math> matrix and dim Nul <math>A=2,</math> then <math>A\vec{x}=\vec{b}</math> is consistent for all <math>\vec{b}</math> in <math>\mathbb{R}^3.</math>
+
'''6.''' True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a &nbsp;<math style="vertical-align: 0px">3\times 5</math>&nbsp; matrix and &nbsp;<math style="vertical-align: -4px">\text{dim Nul }A=2,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">A\vec{x}=\vec{b}</math>&nbsp; is consistent for all &nbsp;<math style="vertical-align: 0px">\vec{b}</math>&nbsp; in &nbsp;<math style="vertical-align: 0px">\mathbb{R}^3.</math>
  
 
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|}
  
−
'''7.''' True or false: Let <math>C=AB</math> for <math>4\times 4</math> matrices <math>A</math> and <math>B.</math> If <math>C</math> is invertible, then <math>A</math> is invertible.
+
'''7.''' True or false: Let &nbsp;<math style="vertical-align: 0px">C=AB</math>&nbsp; for &nbsp;<math style="vertical-align: 0px">4\times 4</math>&nbsp; matrices &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">B.</math>&nbsp; If &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; is invertible, then &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is invertible.
  
 
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−
'''8.''' True or false: Let <math>W</math> be a subspace of <math>\mathbb{R}^4</math> and <math>\vec{v}</math> be a vector in <math>\mathbb{R}^4.</math> If <math>\vec{v}\in W</math> and <math>\vec{v}\in W^\perp,</math> then <math>\vec{v}=\vec{0}.</math>
+
'''8.''' True or false: Let &nbsp;<math style="vertical-align: 0px">W</math>&nbsp; be a subspace of &nbsp;<math style="vertical-align: 0px">\mathbb{R}^4</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; be a vector in &nbsp;<math style="vertical-align: 0px">\mathbb{R}^4.</math>&nbsp; If &nbsp;<math style="vertical-align: 0px">\vec{v}\in W</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">\vec{v}\in W^\perp,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">\vec{v}=\vec{0}.</math>
  
 
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|}
  
−
'''9.''' True or false: If <math>A</math> is an invertible <math>3\times 3</math> matrix, and <math>B</math> and <math>C</math> are <math>3\times 3</math> matrices such that <math>AB=AC,</math> then <math>B=C.</math>
+
'''9.''' True or false: If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is an invertible &nbsp;<math style="vertical-align: 0px">3\times 3</math>&nbsp; matrix, and &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">C</math>&nbsp; are &nbsp;<math style="vertical-align: 0px">3\times 3</math>&nbsp; matrices such that &nbsp;<math style="vertical-align: -4px">AB=AC,</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">B=C.</math>
  
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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'''10.'''  
 
'''10.'''  
  
−
(a) Is the matrix <math>A=     
+
(a) Is the matrix &nbsp;<math style="vertical-align: -18px">A=     
 
     \begin{bmatrix}
 
     \begin{bmatrix}
 
           3 & 1 \\
 
           3 & 1 \\
 
           0 & 3  
 
           0 & 3  
−
         \end{bmatrix}</math> diagonalizable? If so, explain why and diagonalize it. If not, explain why not.
+
         \end{bmatrix}</math>&nbsp; diagonalizable? If so, explain why and diagonalize it. If not, explain why not.
 
          
 
          
−
(b) Is the matrix <math>A=     
+
(b) Is the matrix &nbsp;<math style="vertical-align: -31px">A=     
 
     \begin{bmatrix}
 
     \begin{bmatrix}
 
           2 & 0 & -2 \\
 
           2 & 0 & -2 \\
 
           1 & 3  & 2 \\
 
           1 & 3  & 2 \\
 
           0 & 0 & 3  
 
           0 & 0 & 3  
−
         \end{bmatrix}</math> diagonalizable? If so, explain why and diagonalize it. If not, explain why not.
+
         \end{bmatrix}</math>&nbsp; diagonalizable? If so, explain why and diagonalize it. If not, explain why not.
  
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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|}
 
|}
  
−
'''11.'''  
+
'''11.''' Find the eigenvalues and eigenvectors of the matrix &nbsp;<math style="vertical-align: -31px">A=   
 +
    \begin{bmatrix}
 +
          1 & 1 & 1 \\
 +
          0 & -1  & 1 \\
 +
          0 & 0 & 2
 +
        \end{bmatrix}.</math>
 +
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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|}
 
|}
  
−
'''12.'''  
+
'''12.''' Consider the matrix &nbsp;<math style="vertical-align: -31px">A=   
 +
    \begin{bmatrix}
 +
          1 & -4 & 9 & -7 \\
 +
          -1 & 2  & -4 & 1 \\
 +
          5 & -6 & 10 & 7
 +
        \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
 +
 
 +
::<math>B=   
 +
    \begin{bmatrix}
 +
          1 & 0 & -1 & 5 \\
 +
          0 & -2  & 5 & -6 \\
 +
          0 & 0 & 0 & 0
 +
        \end{bmatrix}.</math>     
 +
   
 +
(a) List rank &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{dim Nul }A.</math>
 +
 
 +
(b) Find bases for &nbsp;<math style="vertical-align: 0px">\text{Col }A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{Nul }A.</math>&nbsp; Find an example of a nonzero vector that belongs to &nbsp;<math style="vertical-align: -5px">\text{Col }A,</math>&nbsp; as well as an example of a nonzero vector that belongs to &nbsp;<math style="vertical-align: 0px">\text{Nul }A.</math>
 +
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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|}
 
|}
  
−
'''13.'''  
+
'''13.''' Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?
 +
 
 +
::<math>\begin{bmatrix}
 +
          1  \\
 +
          0 \\
 +
          2
 +
        \end{bmatrix},
 +
        \begin{bmatrix}
 +
          3  \\
 +
          1 \\
 +
          1
 +
        \end{bmatrix},
 +
        \begin{bmatrix}
 +
          -2  \\
 +
          -1 \\
 +
          1
 +
        \end{bmatrix},
 +
        \begin{bmatrix}
 +
          5  \\
 +
          2 \\
 +
          2
 +
        \end{bmatrix}</math>
 +
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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|}
 
|}
  
−
'''14.'''  
+
'''14.''' Let 
 +
&nbsp;<math>B=   
 +
    \begin{bmatrix}
 +
          1 & -2 & 3 & 4\\
 +
          0 & 3 &0 &0\\
 +
          0 & 5 & 1 & 2\\
 +
          0 & -1 & 3 & 6
 +
        \end{bmatrix}.
 +
</math>
 +
 
 +
(a) Is &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; invertible? Explain.
 +
 
 +
(b) Define a linear transformation &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; by the formula &nbsp;<math style="vertical-align: -5px">T(\vec{x})=B\vec{x}.</math>&nbsp; Is &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; onto? Explain.
 +
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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|}
 
|}
  
−
'''15.'''  
+
'''15.''' Suppose &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation given by the formula
 +
 
 +
::<math>T\Bigg(
 +
\begin{bmatrix}
 +
          x_1 \\
 +
          x_2 \\
 +
          x_3 \\
 +
        \end{bmatrix}
 +
        \Bigg)=
 +
\begin{bmatrix}
 +
          5x_1-2.5x_2+10x_3 \\
 +
          -x_1+0.5x_2-2x_3
 +
        \end{bmatrix}</math>
 +
       
 +
(a) Find the standard matrix for &nbsp;<math style="vertical-align: 0px">T.</math>
 +
       
 +
(b) Let &nbsp;<math style="vertical-align: -5px">\vec{u}=7\vec{e_1}-4\vec{e_2}.</math>&nbsp; Find &nbsp;<math style="vertical-align: -6px">T(\vec{u}).</math>
 +
       
 +
(c) Is &nbsp;<math style="vertical-align: -21px">\begin{bmatrix}
 +
          -1 \\
 +
          3
 +
        \end{bmatrix}</math>&nbsp; in the range of &nbsp;<math style="vertical-align: 0px">T?</math>&nbsp; Explain.
 +
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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|}
 
|}
  
−
'''16.'''  
+
'''16.''' Let &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; be &nbsp;<math style="vertical-align: 0px">6\times 6</math>&nbsp; matrices with &nbsp;<math style="vertical-align: -1px">\text{det }A=-10</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{det }B=5.</math>&nbsp; Use properties of determinants to compute:
 +
 
 +
(a) &nbsp;<math style="vertical-align: -2px">\text{det }3A</math>
 +
 
 +
(b) &nbsp;<math style="vertical-align: -7px">\text{det }(A^TB^{-1})</math>
 +
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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|}
 
|}
  
−
'''17.'''  
+
'''17.''' Let &nbsp;<math style="vertical-align: -20px">A=   
 +
    \begin{bmatrix}
 +
          5 & 1 \\
 +
          0 & 5
 +
        \end{bmatrix}.</math>
 +
 
 +
(a) Find a basis for the eigenspace(s) of &nbsp;<math style="vertical-align: 0px">A.</math>
 +
 
 +
(b) Is the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; diagonalizable? Explain.
 +
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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|}
 
|}
  
−
'''18.'''  
+
'''18.''' Let &nbsp;<math>\vec{v}=\begin{bmatrix}
 +
          -1 \\
 +
          3 \\
 +
          0
 +
        \end{bmatrix}</math>&nbsp; and &nbsp;<math>\vec{y}=\begin{bmatrix}
 +
          2 \\
 +
          0 \\
 +
          5
 +
        \end{bmatrix}.</math>
 +
       
 +
(a) Find a unit vector in the direction of &nbsp;<math style="vertical-align: 0px">\vec{v}.</math>
 +
       
 +
(b) Find the distance between &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; and &nbsp;<math style="vertical-align: -3px">\vec{y}.</math>
 +
       
 +
(c) Let &nbsp;<math style="vertical-align: -5px">L=\text{Span }\{\vec{v}\}.</math>&nbsp; Compute the orthogonal projection of &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; onto &nbsp;<math style="vertical-align: 0px">L.</math>
 +
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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|}
 
|}
  
−
'''19.'''  
+
'''19.''' Let &nbsp;<math>W=\text{Span }\Bigg\{\begin{bmatrix}
 +
          2 \\
 +
          0 \\
 +
          -1 \\
 +
          0
 +
        \end{bmatrix},\begin{bmatrix}
 +
          -3 \\
 +
          1 \\
 +
          0 \\
 +
          0
 +
        \end{bmatrix}\Bigg\}.</math>&nbsp; Is &nbsp;<math>\begin{bmatrix}
 +
          2 \\
 +
          6 \\
 +
          4 \\
 +
          0
 +
        \end{bmatrix}</math>&nbsp; in &nbsp;<math style="vertical-align: 0px">W^\perp?</math>&nbsp; Explain.
 +
 
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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'''20.'''  
 
'''20.'''  
 +
 +
(a) Let &nbsp;<math style="vertical-align: -2px">T:\mathbb{R}^2\rightarrow \mathbb{R}^2</math>&nbsp; be a transformation given by
 +
 +
::<math>T\bigg(
 +
\begin{bmatrix}
 +
          x \\
 +
          y
 +
        \end{bmatrix}
 +
        \bigg)=
 +
\begin{bmatrix}
 +
          1-xy \\
 +
          x+y
 +
        \end{bmatrix}.</math>
 +
 +
Determine whether &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation. Explain.
 +
 +
(b) Let &nbsp;<math style="vertical-align: -19px">A=   
 +
    \begin{bmatrix}
 +
          1 & -3 & 0 \\
 +
          -4 & 1 &1
 +
        \end{bmatrix}</math>&nbsp; and &nbsp;<math style="vertical-align: -32px">B=   
 +
    \begin{bmatrix}
 +
          2 & 1\\
 +
          1 & 0 \\
 +
          -1 & 1
 +
        \end{bmatrix}.</math>&nbsp; Find &nbsp;<math style="vertical-align: -4px">AB,~BA^T</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">A-B^T.</math>
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''21.''' Let &nbsp;<math style="vertical-align: -31px">A=   
 +
    \begin{bmatrix}
 +
          1 & 3 & 8 \\
 +
          2 & 4 &11\\
 +
          1 & 2 & 5
 +
        \end{bmatrix}.</math>&nbsp; Find &nbsp;<math style="vertical-align: 0px">A^{-1}</math>&nbsp; if possible.
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''22.''' Find a formula for &nbsp;<math>\begin{bmatrix}
 +
          1 & -6  \\
 +
          2 & -6
 +
        \end{bmatrix}^k</math>&nbsp; by diagonalizing the matrix.
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''23.'''
 +
 +
(a) Show that if &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; is an eigenvector of the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; corresponding to the eigenvalue 2, then &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; is an eigenvector of &nbsp;<math style="vertical-align: -2px">A^3-A^2+I.</math>&nbsp; What is the corresponding eigenvalue?
 +
 +
(b) Show that if &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; is an eigenvector of the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; corresponding to the eigenvalue 3 and &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is invertible, then &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; is an eigenvector of &nbsp;<math style="vertical-align: 0px">A^{-1}.</math>&nbsp; What is the corresponding eigenvalue?
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''24.''' Let &nbsp;<math>A=\begin{bmatrix}
 +
          3 & 0 & -1 \\
 +
          0 & 1 &-3\\
 +
          1 & 0 & 0
 +
        \end{bmatrix}\begin{bmatrix}
 +
          3 & 0 & 0 \\
 +
          0 & 4 &0\\
 +
          0 & 0 & 3
 +
        \end{bmatrix}\begin{bmatrix}
 +
          0 & 0 & 1 \\
 +
          -3 & 1 &9\\
 +
          -1 & 0 & 3
 +
        \end{bmatrix}.</math>
 +
 +
Use the Diagonalization Theorem to find the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and a basis for each eigenspace.
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''25.''' Give an example of a &nbsp;<math style="vertical-align: 0px">3\times 3</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; with eigenvalues 5,-1 and 3.
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''26.''' Assume &nbsp;<math style="vertical-align: 0px">A^2=I.</math>&nbsp; Find &nbsp;<math style="vertical-align: -1px">\text{Nul }A.</math>
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''27.''' If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is an &nbsp;<math style="vertical-align: 0px">n\times n</math>&nbsp; matrix such that &nbsp;<math style="vertical-align: -4px">AA^T=I,</math>&nbsp; what are the possible values of &nbsp;<math style="vertical-align: 0px">\text{det }A?</math>
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''28.''' Show that if &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; is an eigenvector of the matrix product &nbsp;<math style="vertical-align: 0px">AB</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">B\vec{x}\ne \vec{0},</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">B\vec{x}</math>&nbsp; is an eigenvector of &nbsp;<math style="vertical-align: 0px">BA.</math>
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''29.'''
 +
 +
(a) Suppose a &nbsp;<math style="vertical-align: 0px">6\times 8</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; has 4 pivot columns. What is &nbsp;<math style="vertical-align: -1px">\text{dim Nul }A?</math>&nbsp; Is &nbsp;<math style="vertical-align: -1px">\text{Col }A=\mathbb{R}^4?</math>&nbsp; Why or why not?
 +
 +
(b) If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a &nbsp;<math style="vertical-align: 0px">7\times 5</math>&nbsp; matrix, what is the smallest possible dimension of &nbsp;<math style="vertical-align: -1px">\text{Nul }A?</math>
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''30.''' Consider the following system of equations.
 +
 +
::<math>x_1+kx_2=1</math>
 +
 +
::<math>3x_1+5x_2=2k</math>
 +
 +
Find all real values of &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; such that the system has only one solution.
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''31.''' Suppose &nbsp;<math style="vertical-align: -5px">\{\vec{u},\vec{v}\}</math>&nbsp; is a basis of the eigenspace corresponding to the eigenvalue 0 of a &nbsp;<math style="vertical-align: 0px">5\times 5</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A.</math>
 +
 +
(a) Is &nbsp;<math style="vertical-align: 0px">\vec{w}=\vec{u}-2\vec{v}</math>&nbsp; an eigenvector of &nbsp;<math style="vertical-align: 0px">A?</math>&nbsp; If so, find the corresponding eigenvalue.
 +
 +
If not, explain why.
 +
 +
(b) Find the dimension of &nbsp;<math style="vertical-align: -1px">\text{Col }A.</math>
 +
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;

Latest revision as of 13:11, 25 August 2017

This is a list of sample problems and is meant to represent the material usually covered in Math 31. An actual test may or may not be similar.


1. True or false: If all the entries of a    matrix    are  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7,}   then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }A}   must be  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7^7.}

Solution:  
Final Answer:  

2. True or false: If a matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2}   is diagonalizable, then the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   must be diagonalizable as well.

Solution:  
Final Answer:  

3. True or false: If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is a  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\times 4}   matrix with characteristic equation  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,}   then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is diagonalizable.

Solution:  
Final Answer:  

4. True or false: If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is invertible, then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is diagonalizable.

Solution:  
Final Answer:  

5. True or false: If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}   are invertible  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n}   matrices, then so is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A+B.}

Solution:  
Final Answer:  

6. True or false: If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is a  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 5}   matrix and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{dim Nul }A=2,}   then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\vec{x}=\vec{b}}   is consistent for all  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{b}}   in  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3.}

Solution:  
Final Answer:  

7. True or false: Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C=AB}   for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\times 4}   matrices  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B.}   If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}   is invertible, then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is invertible.

Solution:  
Final Answer:  

8. True or false: Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W}   be a subspace of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^4}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}}   be a vector in  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^4.}   If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\in W}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\in W^\perp,}   then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}=\vec{0}.}

Solution:  
Final Answer:  

9. True or false: If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is an invertible  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3}   matrix, and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C}   are  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3}   matrices such that  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB=AC,}   then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B=C.}

Solution:  
Final Answer:  

10.

(a) Is the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 3 & 1 \\ 0 & 3 \end{bmatrix}}   diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

(b) Is the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 2 & 0 & -2 \\ 1 & 3 & 2 \\ 0 & 0 & 3 \end{bmatrix}}   diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

Solution:  
Final Answer:  

11. Find the eigenvalues and eigenvectors of the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 1 & 1 & 1 \\ 0 & -1 & 1 \\ 0 & 0 & 2 \end{bmatrix}.}

Solution:  
Final Answer:  

12. Consider the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 1 & -4 & 9 & -7 \\ -1 & 2 & -4 & 1 \\ 5 & -6 & 10 & 7 \end{bmatrix}}   and assume that it is row equivalent to the matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B= \begin{bmatrix} 1 & 0 & -1 & 5 \\ 0 & -2 & 5 & -6 \\ 0 & 0 & 0 & 0 \end{bmatrix}.}

(a) List rank  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{dim Nul }A.}

(b) Find bases for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Col }A}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Nul }A.}   Find an example of a nonzero vector that belongs to    as well as an example of a nonzero vector that belongs to  

Solution:  
Final Answer:  

13. Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?

Solution:  
Final Answer:  

14. Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B= \begin{bmatrix} 1 & -2 & 3 & 4\\ 0 & 3 &0 &0\\ 0 & 5 & 1 & 2\\ 0 & -1 & 3 & 6 \end{bmatrix}. }

(a) Is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}   invertible? Explain.

(b) Define a linear transformation  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}   by the formula  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\vec{x})=B\vec{x}.}   Is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}   onto? Explain.

Solution:  
Final Answer:  

15. Suppose  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}   is a linear transformation given by the formula

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\Bigg( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} \Bigg)= \begin{bmatrix} 5x_1-2.5x_2+10x_3 \\ -x_1+0.5x_2-2x_3 \end{bmatrix}}

(a) Find the standard matrix for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T.}

(b) Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}=7\vec{e_1}-4\vec{e_2}.}   Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\vec{u}).}

(c) Is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} -1 \\ 3 \end{bmatrix}}   in the range of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T?}   Explain.

Solution:  
Final Answer:  

16. Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B}   be  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6\times 6}   matrices with  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }A=-10}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }B=5.}   Use properties of determinants to compute:

(a)  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }3A}

(b)  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }(A^TB^{-1})}

Solution:  
Final Answer:  

17. Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 5 & 1 \\ 0 & 5 \end{bmatrix}.}

(a) Find a basis for the eigenspace(s) of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A.}

(b) Is the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   diagonalizable? Explain.

Solution:  
Final Answer:  

18. Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}=\begin{bmatrix} -1 \\ 3 \\ 0 \end{bmatrix}}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}=\begin{bmatrix} 2 \\ 0 \\ 5 \end{bmatrix}.}

(a) Find a unit vector in the direction of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}.}

(b) Find the distance between  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}.}

(c) Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\text{Span }\{\vec{v}\}.}   Compute the orthogonal projection of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}}   onto  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L.}

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19. Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W=\text{Span }\Bigg\{\begin{bmatrix} 2 \\ 0 \\ -1 \\ 0 \end{bmatrix},\begin{bmatrix} -3 \\ 1 \\ 0 \\ 0 \end{bmatrix}\Bigg\}.}   Is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} 2 \\ 6 \\ 4 \\ 0 \end{bmatrix}}   in  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W^\perp?}   Explain.

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20.

(a) Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T:\mathbb{R}^2\rightarrow \mathbb{R}^2}   be a transformation given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\bigg( \begin{bmatrix} x \\ y \end{bmatrix} \bigg)= \begin{bmatrix} 1-xy \\ x+y \end{bmatrix}.}

Determine whether  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T}   is a linear transformation. Explain.

(b) Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 1 & -3 & 0 \\ -4 & 1 &1 \end{bmatrix}}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B= \begin{bmatrix} 2 & 1\\ 1 & 0 \\ -1 & 1 \end{bmatrix}.}   Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB,~BA^T}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A-B^T.}

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21. Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 1 & 3 & 8 \\ 2 & 4 &11\\ 1 & 2 & 5 \end{bmatrix}.}   Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{-1}}   if possible.

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22. Find a formula for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} 1 & -6 \\ 2 & -6 \end{bmatrix}^k}   by diagonalizing the matrix.

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23.

(a) Show that if  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}}   is an eigenvector of the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   corresponding to the eigenvalue 2, then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}}   is an eigenvector of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^3-A^2+I.}   What is the corresponding eigenvalue?

(b) Show that if  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{y}}   is an eigenvector of the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   corresponding to the eigenvalue 3 and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is invertible, then    is an eigenvector of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^{-1}.}   What is the corresponding eigenvalue?

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24. Let  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\begin{bmatrix} 3 & 0 & -1 \\ 0 & 1 &-3\\ 1 & 0 & 0 \end{bmatrix}\begin{bmatrix} 3 & 0 & 0 \\ 0 & 4 &0\\ 0 & 0 & 3 \end{bmatrix}\begin{bmatrix} 0 & 0 & 1 \\ -3 & 1 &9\\ -1 & 0 & 3 \end{bmatrix}.}

Use the Diagonalization Theorem to find the eigenvalues of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   and a basis for each eigenspace.

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25. Give an example of a  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3}   matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   with eigenvalues 5,-1 and 3.

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26. Assume  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2=I.}   Find  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Nul }A.}

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27. If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is an  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n}   matrix such that  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AA^T=I,}   what are the possible values of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{det }A?}

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28. Show that if  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{x}}   is an eigenvector of the matrix product  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\vec{x}\ne \vec{0},}   then  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B\vec{x}}   is an eigenvector of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle BA.}

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29.

(a) Suppose a  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6\times 8}   matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   has 4 pivot columns. What is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{dim Nul }A?}   Is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Col }A=\mathbb{R}^4?}   Why or why not?

(b) If  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is a  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7\times 5}   matrix, what is the smallest possible dimension of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Nul }A?}

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30. Consider the following system of equations.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1+kx_2=1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3x_1+5x_2=2k}

Find all real values of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k}   such that the system has only one solution.

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31. Suppose  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\vec{u},\vec{v}\}}   is a basis of the eigenspace corresponding to the eigenvalue 0 of a  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5\times 5}   matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A.}

(a) Is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w}=\vec{u}-2\vec{v}}   an eigenvector of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A?}   If so, find the corresponding eigenvalue.

If not, explain why.

(b) Find the dimension of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Col }A.}

Solution:  
Final Answer:  
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