Difference between revisions of "031 Review Problems"

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|}
 
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−
'''11.'''  
+
'''11.''' Find the eigenvalues and eigenvectors of the matrix <math>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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|}
 
|}
  
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'''12.'''  
+
'''12.''' Consider the matrix <math>A=   
 +
    \begin{bmatrix}
 +
          1 & -4 & 9 & -7 \\
 +
          -1 & 2  & -4 & 1 \\
 +
          5 & -6 & 10 & 7
 +
        \end{bmatrix}</math> 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 <math>A</math> and dim Nul <math>A.</math>
 +
 
 +
(b) Find bases for Col <math>A</math> and Nul <math>A.</math> Find an example of a nonzero vector that belongs to Col <math>A,</math> as well as an example of a nonzero vector that belongs to Nul <math>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 
 +
<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 <math>B</math> invertible? Explain.
 +
 
 +
(b) Define a linear transformation <math>T</math> by the formula <math>T(\vec{x})=B\vec{x}.</math> Is <math>T</math> 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 <math>T</math> 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 <math>T.</math>
 +
       
 +
(b) Let <math>\vec{u}=7\vec{e_1}-4\vec{e_2}.</math> Find <math>T(\vec{u}).</math>
 +
       
 +
(c) Is <math>\begin{bmatrix}
 +
          -1 \\
 +
          3
 +
        \end{bmatrix}</math> in the range of <math>T?</math> Explain.
 +
 
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;
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'''16.'''  
+
'''16.''' Let <math>A</math> and <math>B</math> be <math>6\times 6</math> matrices with det <math>A=-10</math> and det <math>B=5.</math> Use properties of
 +
 
 +
determinants to compute:
 +
 
 +
(a) det <math>3A</math>
 +
 
 +
(b) det <math>(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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'''20.'''  
 
'''20.'''  
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
 +
|
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''21.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
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|-
 +
|
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|-
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|}
 +
 +
'''22.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
 +
|
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|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
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|-
 +
|
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|-
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|}
 +
 +
'''23.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
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|}
 +
 +
'''24.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''25.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''26.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''27.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
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|-
 +
|
 +
|}
 +
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
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|-
 +
|
 +
|-
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|}
 +
 +
'''28.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
 +
|-
 +
|
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|}
 +
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
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|-
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|
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|-
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|}
 +
 +
'''29.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
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|-
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|
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|}
 +
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
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|-
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|
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|-
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|}
 +
 +
'''30.'''
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Solution: &nbsp;
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|-
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|
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|}
 +
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Final Answer: &nbsp;
 +
|-
 +
|
 +
|-
 +
|}
 +
 +
'''31.'''
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Solution: &nbsp;
 
!Solution: &nbsp;

Revision as of 16:53, 24 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 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 7} 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} 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 det 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 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 dim Nul 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,} 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 dim Nul 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) Find bases for Col and Nul Find an example of a nonzero vector that belongs to Col as well as an example of a nonzero vector that belongs to Nul

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Final Answer:  

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

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14. Let

(a) Is invertible? Explain.

(b) Define a linear transformation by the formula Is onto? Explain.

Solution:  
Final Answer:  

15. Suppose is a linear transformation given by the formula

(a) Find the standard matrix for

(b) Let Find

(c) Is in the range of Explain.

Solution:  
Final Answer:  

16. Let and be matrices with det and det Use properties of

determinants to compute:

(a) det

(b) det

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17.

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18.

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19.

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20.

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21.

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22.

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23.

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24.

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25.

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26.

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27.

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28.

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29.

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30.

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31.

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