Difference between revisions of "031 Review Part 3"
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| − | ''' | + | '''These questions are from sample exams and actual exams at other universities. The questions are meant to represent the material usually covered in Math 31 for the final. An actual test may or may not be similar.''' |
'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.''' | '''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.''' | ||
<div class="noautonum">__TOC__</div> | <div class="noautonum">__TOC__</div> | ||
| − | == [[ | + | == [[031_Review Part 3,_Problem_1|<span class="biglink"><span style="font-size:80%"> Problem 1 </span></span>]]== |
<span class="exam">(a) Is the matrix <math style="vertical-align: -18px">A= | <span class="exam">(a) Is the matrix <math style="vertical-align: -18px">A= | ||
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\end{bmatrix}</math> diagonalizable? If so, explain why and diagonalize it. If not, explain why not. | \end{bmatrix}</math> diagonalizable? If so, explain why and diagonalize it. If not, explain why not. | ||
| − | + | == [[031_Review Part 3,_Problem_2|<span class="biglink"><span style="font-size:80%"> Problem 2 </span>]] == | |
| − | |||
| − | == [[ | ||
<span class="exam"> Find the eigenvalues and eigenvectors of the matrix <math style="vertical-align: -31px">A= | <span class="exam"> Find the eigenvalues and eigenvectors of the matrix <math style="vertical-align: -31px">A= | ||
\begin{bmatrix} | \begin{bmatrix} | ||
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\end{bmatrix}.</math> | \end{bmatrix}.</math> | ||
| − | + | == [[031_Review Part 3,_Problem_3|<span class="biglink"><span style="font-size:80%"> Problem 3 </span>]] == | |
| − | == [[ | ||
<span class="exam">Let <math style="vertical-align: -20px">A= | <span class="exam">Let <math style="vertical-align: -20px">A= | ||
\begin{bmatrix} | \begin{bmatrix} | ||
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<span class="exam">(b) Is the matrix <math style="vertical-align: 0px">A</math> diagonalizable? Explain. | <span class="exam">(b) Is the matrix <math style="vertical-align: 0px">A</math> diagonalizable? Explain. | ||
| − | == [[ | + | == [[031_Review Part 3,_Problem_4|<span class="biglink"><span style="font-size:80%"> Problem 4 </span>]] == |
<span class="exam"> Let <math>W=\text{Span }\Bigg\{\begin{bmatrix} | <span class="exam"> Let <math>W=\text{Span }\Bigg\{\begin{bmatrix} | ||
2 \\ | 2 \\ | ||
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\end{bmatrix}</math> in <math style="vertical-align: 0px">W^\perp?</math> Explain. | \end{bmatrix}</math> in <math style="vertical-align: 0px">W^\perp?</math> Explain. | ||
| − | + | == [[031_Review Part 3,_Problem_5|<span class="biglink"><span style="font-size:80%"> Problem 5 </span>]] == | |
| − | == [[ | ||
<span class="exam">Find a formula for <math>\begin{bmatrix} | <span class="exam">Find a formula for <math>\begin{bmatrix} | ||
1 & -6 \\ | 1 & -6 \\ | ||
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\end{bmatrix}^k</math> by diagonalizing the matrix. | \end{bmatrix}^k</math> by diagonalizing the matrix. | ||
| − | == [[ | + | == [[031_Review Part 3,_Problem_6|<span class="biglink"><span style="font-size:80%"> Problem 6 </span>]] == |
<span class="exam"> (a) Show that if <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of the matrix <math style="vertical-align: 0px">A</math> corresponding to the eigenvalue 2, then <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of <math style="vertical-align: -2px">A^3-A^2+I.</math> What is the corresponding eigenvalue? | <span class="exam"> (a) Show that if <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of the matrix <math style="vertical-align: 0px">A</math> corresponding to the eigenvalue 2, then <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of <math style="vertical-align: -2px">A^3-A^2+I.</math> What is the corresponding eigenvalue? | ||
<span class="exam">(b) Show that if <math style="vertical-align: -3px">\vec{y}</math> is an eigenvector of the matrix <math style="vertical-align: 0px">A</math> corresponding to the eigenvalue 3 and <math style="vertical-align: 0px">A</math> is invertible, then <math style="vertical-align: -3px">\vec{y}</math> is an eigenvector of <math style="vertical-align: 0px">A^{-1}.</math> What is the corresponding eigenvalue? | <span class="exam">(b) Show that if <math style="vertical-align: -3px">\vec{y}</math> is an eigenvector of the matrix <math style="vertical-align: 0px">A</math> corresponding to the eigenvalue 3 and <math style="vertical-align: 0px">A</math> is invertible, then <math style="vertical-align: -3px">\vec{y}</math> is an eigenvector of <math style="vertical-align: 0px">A^{-1}.</math> What is the corresponding eigenvalue? | ||
| − | == [[ | + | == [[031_Review Part 3,_Problem_7|<span class="biglink"><span style="font-size:80%"> Problem 7 </span>]] == |
<span class="exam">Let <math>A=\begin{bmatrix} | <span class="exam">Let <math>A=\begin{bmatrix} | ||
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<span class="exam">Use the Diagonalization Theorem to find the eigenvalues of <math style="vertical-align: 0px">A</math> and a basis for each eigenspace. | <span class="exam">Use the Diagonalization Theorem to find the eigenvalues of <math style="vertical-align: 0px">A</math> and a basis for each eigenspace. | ||
| − | == [[ | + | == [[031_Review Part 3,_Problem_8|<span class="biglink"><span style="font-size:80%"> Problem 8 </span>]] == |
<span class="exam">Give an example of a <math style="vertical-align: 0px">3\times 3</math> matrix <math style="vertical-align: 0px">A</math> with eigenvalues 5,-1 and 3. | <span class="exam">Give an example of a <math style="vertical-align: 0px">3\times 3</math> matrix <math style="vertical-align: 0px">A</math> with eigenvalues 5,-1 and 3. | ||
| − | == [[ | + | == [[031_Review Part 3,_Problem_9|<span class="biglink"><span style="font-size:80%"> Problem 9 </span>]] == |
<span class="exam">Assume <math style="vertical-align: 0px">A^2=I.</math> Find <math style="vertical-align: -1px">\text{Nul }A.</math> | <span class="exam">Assume <math style="vertical-align: 0px">A^2=I.</math> Find <math style="vertical-align: -1px">\text{Nul }A.</math> | ||
| − | + | == [[031_Review Part 3,_Problem_10|<span class="biglink"><span style="font-size:80%"> Problem 10 </span>]] == | |
| − | == [[ | ||
<span class="exam">Show that if <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of the matrix product <math style="vertical-align: 0px">AB</math> and <math style="vertical-align: -5px">B\vec{x}\ne \vec{0},</math> then <math style="vertical-align: 0px">B\vec{x}</math> is an eigenvector of <math style="vertical-align: 0px">BA.</math> | <span class="exam">Show that if <math style="vertical-align: 0px">\vec{x}</math> is an eigenvector of the matrix product <math style="vertical-align: 0px">AB</math> and <math style="vertical-align: -5px">B\vec{x}\ne \vec{0},</math> then <math style="vertical-align: 0px">B\vec{x}</math> is an eigenvector of <math style="vertical-align: 0px">BA.</math> | ||
| − | == [[ | + | == [[031_Review Part 3,_Problem_11|<span class="biglink"><span style="font-size:80%"> Problem 11 </span>]] == |
<span class="exam">Suppose <math style="vertical-align: -5px">\{\vec{u},\vec{v}\}</math> is a basis of the eigenspace corresponding to the eigenvalue 0 of a <math style="vertical-align: 0px">5\times 5</math> matrix <math style="vertical-align: 0px">A.</math> | <span class="exam">Suppose <math style="vertical-align: -5px">\{\vec{u},\vec{v}\}</math> is a basis of the eigenspace corresponding to the eigenvalue 0 of a <math style="vertical-align: 0px">5\times 5</math> matrix <math style="vertical-align: 0px">A.</math> | ||
Latest revision as of 18:34, 9 October 2017
These questions are from sample exams and actual exams at other universities. The questions are meant to represent the material usually covered in Math 31 for the final. An actual test may or may not be similar.
Click on the boxed problem numbers to go to a solution.
Problem 1
(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.
Problem 2
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}.}
Problem 3
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.
Problem 4
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.
Problem 5
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.
Problem 6
(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 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 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?
Problem 7
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.
Problem 8
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.
Problem 9
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.}
Problem 10
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.}
Problem 11
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.}