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 | + | '''1.''' True or false: If all the entries of a <math>7\times 7</math> matrix <math>A</math> are <math>7,</math> then <math>\text{det }A</math> must be <math>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 <math>A^2</math> is diagonalizable, then the matrix <math>A</math> 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 <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. |
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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 <math>A</math> is invertible, then <math>A</math> 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 <math>A</math> and <math>B</math> are invertible <math>n\times n</math> matrices, then so is <math>A+B.</math> |
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| − | '''6.''' True or false: If <math>A</math> is a <math>3\times 5</math> matrix and | + | '''6.''' True or false: If <math>A</math> is a <math>3\times 5</math> matrix and <math>\text{dim Nul }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> |
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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 <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. |
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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 <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> |
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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 <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> |
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'''10.''' | '''10.''' | ||
| − | (a) Is the matrix <math>A= | + | (a) Is the matrix <math>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> diagonalizable? If so, explain why and diagonalize it. If not, explain why not. |
| − | (b) Is the matrix <math>A= | + | (b) Is the matrix <math>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> diagonalizable? If so, explain why and diagonalize it. If not, explain why not. |
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|} | |} | ||
| − | '''11.''' Find the eigenvalues and eigenvectors of the matrix <math>A= | + | '''11.''' Find the eigenvalues and eigenvectors of the matrix <math>A= |
\begin{bmatrix} | \begin{bmatrix} | ||
1 & 1 & 1 \\ | 1 & 1 & 1 \\ | ||
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|} | |} | ||
| − | '''12.''' Consider the matrix <math>A= | + | '''12.''' Consider the matrix <math>A= |
\begin{bmatrix} | \begin{bmatrix} | ||
1 & -4 & 9 & -7 \\ | 1 & -4 & 9 & -7 \\ | ||
-1 & 2 & -4 & 1 \\ | -1 & 2 & -4 & 1 \\ | ||
5 & -6 & 10 & 7 | 5 & -6 & 10 & 7 | ||
| − | \end{bmatrix}</math> and assume that it is row equivalent to the matrix | + | \end{bmatrix}</math> and assume that it is row equivalent to the matrix |
| − | <math>B= | + | ::<math>B= |
\begin{bmatrix} | \begin{bmatrix} | ||
1 & 0 & -1 & 5 \\ | 1 & 0 & -1 & 5 \\ | ||
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\end{bmatrix}.</math> | \end{bmatrix}.</math> | ||
| − | (a) List rank <math>A</math> and | + | (a) List rank <math>A</math> and <math>\text{dim Nul }A.</math> |
| − | (b) Find bases for | + | (b) Find bases for <math>\text{Col }A</math> and <math>\text{Nul }A.</math> Find an example of a nonzero vector that belongs to <math>\text{Col }A,</math> as well as an example of a nonzero vector that belongs to <math>\text{Nul }A.</math> |
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'''13.''' Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent? | '''13.''' Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent? | ||
| − | <math>\begin{bmatrix} | + | ::<math>\begin{bmatrix} |
1 \\ | 1 \\ | ||
0 \\ | 0 \\ | ||
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'''14.''' Let | '''14.''' Let | ||
| − | <math>B= | + | <math>B= |
\begin{bmatrix} | \begin{bmatrix} | ||
1 & -2 & 3 & 4\\ | 1 & -2 & 3 & 4\\ | ||
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</math> | </math> | ||
| − | (a) Is <math>B</math> invertible? Explain. | + | (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. | + | (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. |
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|} | |} | ||
| − | '''15.''' Suppose <math>T</math> is a linear transformation given by the formula | + | '''15.''' Suppose <math>T</math> is a linear transformation given by the formula |
| − | <math>T\Bigg( | + | ::<math>T\Bigg( |
\begin{bmatrix} | \begin{bmatrix} | ||
x_1 \\ | x_1 \\ | ||
| Line 293: | Line 293: | ||
\end{bmatrix}</math> | \end{bmatrix}</math> | ||
| − | (a) Find the standard matrix for <math>T.</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> | + | (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} | + | (c) Is <math>\begin{bmatrix} |
-1 \\ | -1 \\ | ||
3 | 3 | ||
| − | \end{bmatrix}</math> in the range of <math>T?</math> Explain. | + | \end{bmatrix}</math> in the range of <math>T?</math> Explain. |
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|} | |} | ||
| − | '''16.''' Let <math>A</math> and <math>B</math> be <math>6\times 6</math> matrices with | + | '''16.''' Let <math>A</math> and <math>B</math> be <math>6\times 6</math> matrices with <math>\text{det }A=-10</math> and <math>\text{det }B=5.</math> Use properties of determinants to compute: |
| − | + | (a) <math>\text{det }3A</math> | |
| − | ( | + | (b) <math>\text{det }(A^TB^{-1})</math> |
| − | |||
| − | |||
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| − | '''17.''' Let <math>A= | + | '''17.''' Let <math>A= |
\begin{bmatrix} | \begin{bmatrix} | ||
5 & 1 \\ | 5 & 1 \\ | ||
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\end{bmatrix}.</math> | \end{bmatrix}.</math> | ||
| − | (a) Find a basis for the eigenspace(s) of <math>A.</math> | + | (a) Find a basis for the eigenspace(s) of <math>A.</math> |
| − | (b) Is the matrix <math>A</math> diagonalizable? Explain. | + | (b) Is the matrix <math>A</math> diagonalizable? Explain. |
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|} | |} | ||
| − | '''18.''' Let <math>\vec{v}=\begin{bmatrix} | + | '''18.''' Let <math>\vec{v}=\begin{bmatrix} |
-1 \\ | -1 \\ | ||
3 \\ | 3 \\ | ||
0 | 0 | ||
| − | \end{bmatrix}</math> and <math>\vec{y}=\begin{bmatrix} | + | \end{bmatrix}</math> and <math>\vec{y}=\begin{bmatrix} |
2 \\ | 2 \\ | ||
0 \\ | 0 \\ | ||
| Line 369: | Line 367: | ||
\end{bmatrix}.</math> | \end{bmatrix}.</math> | ||
| − | (a) Find a unit vector in the direction of <math>\vec{v}.</math> | + | (a) Find a unit vector in the direction of <math>\vec{v}.</math> |
| − | (b) Find the distance between <math>\vec{v}</math> and <math>\vec{y}.</math> | + | (b) Find the distance between <math>\vec{v}</math> and <math>\vec{y}.</math> |
| − | (c) Let <math>L= | + | (c) Let <math>L=\text{Span }\{\vec{v}\}.</math> Compute the orthogonal projection of <math>\vec{y}</math> onto <math>L.</math> |
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|} | |} | ||
| − | '''19.''' Let <math>W= | + | '''19.''' Let <math>W=\text{Span }\Bigg\{\begin{bmatrix} |
2 \\ | 2 \\ | ||
0 \\ | 0 \\ | ||
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0 \\ | 0 \\ | ||
0 | 0 | ||
| − | \end{bmatrix}\Bigg\}.</math> Is <math>\begin{bmatrix} | + | \end{bmatrix}\Bigg\}.</math> Is <math>\begin{bmatrix} |
2 \\ | 2 \\ | ||
6 \\ | 6 \\ | ||
4 \\ | 4 \\ | ||
0 | 0 | ||
| − | \end{bmatrix}</math> in <math>W^\perp?</math> Explain. | + | \end{bmatrix}</math> in <math>W^\perp?</math> Explain. |
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'''20.''' | '''20.''' | ||
| − | (a) Let <math>T:\mathbb{R}^2\rightarrow \mathbb{R}^2</math> be a transformation given by | + | (a) Let <math>T:\mathbb{R}^2\rightarrow \mathbb{R}^2</math> be a transformation given by |
| − | <math>T\bigg( | + | ::<math>T\bigg( |
\begin{bmatrix} | \begin{bmatrix} | ||
x \\ | x \\ | ||
| Line 433: | Line 431: | ||
\end{bmatrix}.</math> | \end{bmatrix}.</math> | ||
| − | Determine whether <math>T</math> is a linear transformation. Explain. | + | Determine whether <math>T</math> is a linear transformation. Explain. |
| − | (b) Let <math>A= | + | (b) Let <math>A= |
\begin{bmatrix} | \begin{bmatrix} | ||
1 & -3 & 0 \\ | 1 & -3 & 0 \\ | ||
-4 & 1 &1 | -4 & 1 &1 | ||
| − | \end{bmatrix}</math> and <math>B= | + | \end{bmatrix}</math> and <math>B= |
\begin{bmatrix} | \begin{bmatrix} | ||
2 & 1\\ | 2 & 1\\ | ||
1 & 0 \\ | 1 & 0 \\ | ||
-1 & 1 | -1 & 1 | ||
| − | \end{bmatrix}.</math> Find <math>AB,</math> <math>BA^T</math> and <math>A-B^T.</math> | + | \end{bmatrix}.</math> Find <math>AB,</math> <math>BA^T</math> and <math>A-B^T.</math> |
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|} | |} | ||
| − | '''21.''' Let <math>A= | + | '''21.''' Let <math>A= |
\begin{bmatrix} | \begin{bmatrix} | ||
1 & 3 & 8 \\ | 1 & 3 & 8 \\ | ||
2 & 4 &11\\ | 2 & 4 &11\\ | ||
1 & 2 & 5 | 1 & 2 & 5 | ||
| − | \end{bmatrix}.</math> Find <math>A^{-1}</math> if possible. | + | \end{bmatrix}.</math> Find <math>A^{-1}</math> if possible. |
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|} | |} | ||
| − | '''22.''' Find a formula for <math>\begin{bmatrix} | + | '''22.''' Find a formula for <math>\begin{bmatrix} |
1 & -6 \\ | 1 & -6 \\ | ||
2 & -6 | 2 & -6 | ||
| − | \end{bmatrix}^k</math> by diagonalizing the matrix. | + | \end{bmatrix}^k</math> by diagonalizing the matrix. |
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'''23.''' | '''23.''' | ||
| − | (a) Show that if <math>\vec{x}</math> is an eigenvector of the matrix <math>A</math> corresponding to the eigenvalue 2, then <math>\vec{x}</math> is an eigenvector of <math>A^3-A^2+I.</math> What is the corresponding eigenvalue? | + | (a) Show that if <math>\vec{x}</math> is an eigenvector of the matrix <math>A</math> corresponding to the eigenvalue 2, then <math>\vec{x}</math> is an eigenvector of <math>A^3-A^2+I.</math> What is the corresponding eigenvalue? |
| − | (b) Show that if <math>\vec{y}</math> is an eigenvector of the matrix <math>A</math> corresponding to the eigenvalue 3 and <math>A</math> is invertible, then <math>\vec{y}</math> is an eigenvector of <math>A^{-1}.</math> What is the corresponding eigenvalue? | + | (b) Show that if <math>\vec{y}</math> is an eigenvector of the matrix <math>A</math> corresponding to the eigenvalue 3 and <math>A</math> is invertible, then <math>\vec{y}</math> is an eigenvector of <math>A^{-1}.</math> What is the corresponding eigenvalue? |
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|} | |} | ||
| − | '''24.''' Let <math>A=\begin{bmatrix} | + | '''24.''' Let <math>A=\begin{bmatrix} |
3 & 0 & -1 \\ | 3 & 0 & -1 \\ | ||
0 & 1 &-3\\ | 0 & 1 &-3\\ | ||
| Line 530: | Line 528: | ||
\end{bmatrix}.</math> | \end{bmatrix}.</math> | ||
| − | Use the Diagonalization Theorem to find the eigenvalues of <math>A</math> and a basis for each eigenspace. | + | Use the Diagonalization Theorem to find the eigenvalues of <math>A</math> and a basis for each eigenspace. |
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|} | |} | ||
| − | '''25.''' Give an example of a <math>3\times 3</math> matrix <math>A</math> with eigenvalues 5,-1 and 3. | + | '''25.''' Give an example of a <math>3\times 3</math> matrix <math>A</math> with eigenvalues 5,-1 and 3. |
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|} | |} | ||
| − | '''26.''' Assume <math>A^2=I.</math> Find | + | '''26.''' Assume <math>A^2=I.</math> Find <math>\text{Nul }A.</math> |
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|} | |} | ||
| − | '''27.''' If <math>A</math> is an <math>n\times n</math> matrix such that <math>AA^T=I,</math> what are the possible values of | + | '''27.''' If <math>A</math> is an <math>n\times n</math> matrix such that <math>AA^T=I,</math> what are the possible values of <math>\text{det }A?</math> |
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|} | |} | ||
| − | '''28.''' Show that if <math>\vec{x}</math> is an eigenvector of the matrix product <math>AB</math> and <math>B\vec{x}\ne \vec{0},</math> then <math>B\vec{x}</math> is an eigenvector of <math>BA.</math> | + | '''28.''' Show that if <math>\vec{x}</math> is an eigenvector of the matrix product <math>AB</math> and <math>B\vec{x}\ne \vec{0},</math> then <math>B\vec{x}</math> is an eigenvector of <math>BA.</math> |
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'''29.''' | '''29.''' | ||
| − | (a) Suppose a <math>6\times 8</math> matrix <math>A</math> has 4 pivot columns. What is | + | (a) Suppose a <math>6\times 8</math> matrix <math>A</math> has 4 pivot columns. What is <math>\text{dim Nul }A?</math> Is <math>\text{Col }A=\mathbb{R}^4?</math> Why or why not? |
| − | (b) If <math>A</math> is a <math>7\times 5</math> matrix, what is the smallest possible dimension of | + | (b) If <math>A</math> is a <math>7\times 5</math> matrix, what is the smallest possible dimension of <math>\text{Nul }A?</math> |
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'''30.''' Consider the following system of equations. | '''30.''' Consider the following system of equations. | ||
| − | <math>x_1+kx_2=1</math> | + | ::<math>x_1+kx_2=1</math> |
| − | <math>3x_1+5x_2=2k</math> | + | ::<math>3x_1+5x_2=2k</math> |
| − | Find all real values of <math>k</math> such that the system has only one solution. | + | Find all real values of <math>k</math> such that the system has only one solution. |
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|} | |} | ||
| − | '''31.''' Suppose <math>\{\vec{u},\vec{v}\}</math> is a basis of the eigenspace corresponding to the eigenvalue 0 of a <math>5\times 5</math> matrix <math>A.</math> | + | '''31.''' Suppose <math>\{\vec{u},\vec{v}\}</math> is a basis of the eigenspace corresponding to the eigenvalue 0 of a <math>5\times 5</math> matrix <math>A.</math> |
| − | (a) Is <math>\vec{w}=\vec{u}-2\vec{v}</math> an eigenvector of <math>A?</math> If so, find the corresponding eigenvalue. | + | (a) Is <math>\vec{w}=\vec{u}-2\vec{v}</math> an eigenvector of <math>A?</math> If so, find the corresponding eigenvalue. |
If not, explain why. | If not, explain why. | ||
| − | (b) Find the dimension of | + | (b) Find the dimension of <math>\text{Col }A.</math> |
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Revision as of 10:33, 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 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 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.}
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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.
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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.
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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.
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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.}
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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.}
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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.
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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 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}.}
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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.}
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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.
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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}.}
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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 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,} as well as an example of a nonzero vector that belongs to 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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13. Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?
- 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 \\ 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}}
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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.
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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.
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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})}
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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.
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18. Let 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,} 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^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 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?
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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.}
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