Difference between revisions of "031 Review Part 3, Problem 7"

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(Created page with "<span class="exam">Consider the matrix  <math style="vertical-align: -31px">A= \begin{bmatrix} 1 & -4 & 9 & -7 \\ -1 & 2 & -4 & 1 \\...")
 
 
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<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=  
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<span class="exam">Let &nbsp;<math>A=\begin{bmatrix}
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    \begin{bmatrix}
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           3 & 0 & -1 \\
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           1 & -4 & 9 & -7 \\
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           0 & 1 &-3\\
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           -1 & 2  & -4 & 1 \\
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           1 & 0 & 0
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           5 & -6 & 10 & 7
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         \end{bmatrix}\begin{bmatrix}
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         \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
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           3 & 0 & 0 \\
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           0 & 4 &0\\
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::<math>B=   
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           0 & 0 & 3
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    \begin{bmatrix}
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         \end{bmatrix}\begin{bmatrix}
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           1 & 0 & -1 & 5 \\
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          0 & 0 & 1 \\
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           0 & -2  & 5 & -6 \\
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          -3 & 1 &9\\
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           0 & 0 & 0 & 0
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          -1 & 0 & 3
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         \end{bmatrix}.</math>     
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        \end{bmatrix}.</math>  
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<span class="exam">(a) List rank &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{dim Nul }A.</math>
 
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<span class="exam">(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>
 
  
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<span class="exam">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;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 
|-
 
|-
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|
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|'''Diagonalization Theorem'''
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|-
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|An &nbsp;<math style="vertical-align: 0px">n\times n</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is diagonalizable if and only if &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; has &nbsp;<math style="vertical-align: 0px">n</math>&nbsp; linearly independent eigenvectors.
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|-
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|In fact, &nbsp;<math style="vertical-align: -4px">A=PDP^{-1},</math>&nbsp; with &nbsp;<math style="vertical-align: 0px">D</math>&nbsp; a diagonal matrix, if and only if the columns of &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; are &nbsp;<math style="vertical-align: 0px">n</math>&nbsp; linearly
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|-
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|independent eigenvectors of &nbsp;<math style="vertical-align: 0px">A.</math>&nbsp; In this case, the diagonal entries of &nbsp;<math style="vertical-align: 0px">D</math>&nbsp; are eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; that
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|-
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|correspond, respectively , to the eigenvectors in &nbsp;<math style="vertical-align: 0px">P.</math>
 
|}
 
|}
  
  
 
'''Solution:'''
 
'''Solution:'''
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'''(a)'''
 
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
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|-
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|Since
 
|-
 
|-
 
|
 
|
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::<math>\begin{bmatrix}
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          3 & 0 & 0 \\
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          0 & 4 &0\\
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          0 & 0 & 3
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        \end{bmatrix}</math>
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|-
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|is a diagonal matrix, the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; and &nbsp;<math style="vertical-align: -1px">4</math>&nbsp; by the Diagonalization Theorem.
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
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|-
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|By the Diagonalization Theorem, a basis for the eigenspace corresponding
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|-
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|to the eigenvalue &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; is
 
|-
 
|-
 
|
 
|
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|}
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::<math>\Bigg\{\begin{bmatrix}
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          3 \\
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'''(b)'''
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          0 \\
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          1
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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        \end{bmatrix},\begin{bmatrix}
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!Step 1: &nbsp;
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          -1 \\
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          -3\\
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          0
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        \end{bmatrix}\Bigg\}</math>
 
|-
 
|-
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|
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|and a basis for the eigenspace corresponding to the eigenvalue &nbsp;<math style="vertical-align: -1px">4</math>&nbsp; is
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|}
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 2: &nbsp;
 
 
|-
 
|-
 
|
 
|
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::<math>\Bigg\{\begin{bmatrix}
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          0 \\
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          1 \\
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          1
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        \end{bmatrix}\Bigg\}.</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
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|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
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|&nbsp; &nbsp; &nbsp; &nbsp; The eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; and &nbsp;<math style="vertical-align: -1px">4.</math>&nbsp;
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; A basis for the eigenspace corresponding
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; to the eigenvalue &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; is
 
|-
 
|-
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|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
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|
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::::<math>\Bigg\{\begin{bmatrix}
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          3 \\
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          0 \\
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          1
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        \end{bmatrix},\begin{bmatrix}
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          -1 \\
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          -3\\
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          0
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        \end{bmatrix}\Bigg\}</math>
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|-
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|&nbsp; &nbsp; &nbsp; &nbsp; and a basis for the eigenspace corresponding to the eigenvalue &nbsp;<math style="vertical-align: -1px">4</math>&nbsp; is
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|-
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|
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::::<math>\Bigg\{\begin{bmatrix}
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          0 \\
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          1 \\
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          1
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        \end{bmatrix}\Bigg\}.</math>
 
|}
 
|}
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[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
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[[031_Review_Part_3|'''<u>Return to Review Problems</u>''']]

Latest revision as of 13:03, 15 October 2017

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.

Foundations:  
Diagonalization Theorem
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  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 if and only 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}   has  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}   linearly independent eigenvectors.
In fact,  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=PDP^{-1},}   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 D}   a diagonal matrix, if and only if the columns 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 P}   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 n}   linearly
independent eigenvectors 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.}   In this case, the diagonal entries 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 D}   are 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}   that
correspond, respectively , to the eigenvectors 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 P.}


Solution:

Step 1:  
Since
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}3&0&0\\0&4&0\\0&0&3\end{bmatrix}}}
is a diagonal matrix, 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}   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}   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 4}   by the Diagonalization Theorem.
Step 2:  
By the Diagonalization Theorem, a basis for the eigenspace corresponding
to the eigenvalue  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}   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 \Bigg\{\begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix},\begin{bmatrix} -1 \\ -3\\ 0 \end{bmatrix}\Bigg\}}
and a basis for the eigenspace corresponding to the eigenvalue  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}   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 \Bigg\{\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}\Bigg\}.}


Final Answer:  
        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}   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}   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 4.}  
        A basis for the eigenspace corresponding
        to the eigenvalue  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}   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 \Bigg\{\begin{bmatrix} 3 \\ 0 \\ 1 \end{bmatrix},\begin{bmatrix} -1 \\ -3\\ 0 \end{bmatrix}\Bigg\}}
        and a basis for the eigenspace corresponding to the eigenvalue  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}   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 \Bigg\{\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}\Bigg\}.}

Return to Review Problems

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