Difference between revisions of "031 Review Problems"

Revision as of 17:00, 24 August 2017

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

1. True or false: If all the entries of a Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 7\times 7} matrix $A$ are $7,$ then det $A$ must be Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 7^{7}.}

Solution:

Final Answer:

2. True or false: If a matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A^{2}} is diagonalizable, then the matrix $A$ must be diagonalizable as well.

Solution:

Final Answer:

3. True or false: If $A$ is a Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 4\times 4} matrix with characteristic equation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lambda (\lambda -1)(\lambda +1)(\lambda +e)=0,} then $A$ is diagonalizable.

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4. True or false: If $A$ is invertible, then $A$ is diagonalizable.

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5. True or false: If $A$ and $B$ are invertible $n\times n$ matrices, then so is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A+B.}

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6. True or false: If $A$ is a $3\times 5$ matrix and dim Nul $A=2,$ then $A{\vec {x}}={\vec {b}}$ is consistent for all ${\vec {b}}$ in $\mathbb {R} ^{3}.$

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7. True or false: Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle C=AB} for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 4\times 4} matrices $A$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B.} If $C$ is invertible, then $A$ is invertible.

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8. True or false: Let $W$ be a subspace of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} ^{4}} and ${\vec {v}}$ be a vector in $\mathbb {R} ^{4}.$ If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}\in W} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}\in W^{\perp },} then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}={\vec {0}}.}

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9. True or false: If $A$ is an invertible $3\times 3$ matrix, and $B$ and $C$ are $3\times 3$ matrices such that $AB=AC,$ then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B=C.}

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10.

(a) Is the matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B={\begin{bmatrix}1&0&-1&5\\0&-2&5&-6\\0&0&0&0\end{bmatrix}}.}

(a) List rank $A$ and dim Nul $A.$

(b) Find bases for Col $A$ and Nul $A.$ Find an example of a nonzero vector that belongs to Col $A,$ as well as an example of a nonzero vector that belongs to Nul $A.$

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

${\begin{bmatrix}1\\0\\2\end{bmatrix}},{\begin{bmatrix}3\\1\\1\end{bmatrix}},{\begin{bmatrix}-2\\-1\\1\end{bmatrix}},{\begin{bmatrix}5\\2\\2\end{bmatrix}}$

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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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Final Answer:

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

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

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

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

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

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Final Answer:

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

determinants to compute:

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

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

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Final Answer:

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

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

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

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Final Answer:

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

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

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

(c) Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=} SpanFailed to parse (MathML with SVG or PNG fallback (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}\}.} 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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Final Answer:

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=} SpanFailed to parse (MathML with SVG or PNG fallback (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} 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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Final Answer:

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.}

Solution:

Final Answer:

21.

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22.

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23.

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24.

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25.

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26.

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27.

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28.

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31.

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@@ Line 336: / Line 336: @@
 |}
-'''17.'''
+'''17.''' Let <math>A=
+    \begin{bmatrix}
+& 1 \\
+& 5
+         \end{bmatrix}.</math>
+(a) Find a basis for the eigenspace(s) of <math>A.</math>
+(b) Is the matrix <math>A</math> diagonalizable? Explain.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 350: / Line 359: @@
 |}
-'''18.'''
+'''18.''' Let <math>\vec{v}=\begin{bmatrix}
+           -1 \\
+\\
+         \end{bmatrix}</math> and <math>\vec{y}=\begin{bmatrix}
+\\
+\\
+         \end{bmatrix}.</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>
+(c) Let <math>L=</math>Span<math>\{\vec{v}\}.</math> Compute the orthogonal projection of <math>\vec{y}</math> onto <math>L.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 364: / Line 388: @@
 |}
-'''19.'''
+'''19.''' Let <math>W=</math>Span<math>\Bigg\{\begin{bmatrix}
+\\
+\\
+           -1 \\
+         \end{bmatrix},\begin{bmatrix}
+           -3 \\
+\\
+\\
+         \end{bmatrix}\Bigg\}.</math> Is <math>\begin{bmatrix}
+\\
+\\
+\\
+         \end{bmatrix}</math> in <math>W^\perp?</math> Explain.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 379: / Line 419: @@
 '''20.'''
+(a) Let <math>T:\mathbb{R}^2\rightarrow \mathbb{R}^2</math> be a transformation given by
+<math>T\bigg(
+\begin{bmatrix}
+           x \\
+           y
+         \end{bmatrix}
+         \bigg)=
+ \begin{bmatrix}
+-xy \\
+           x+y
+         \end{bmatrix}.</math>
+Determine whether <math>T</math> is a linear transformation. Explain.
+(b) Let <math>A=
+    \begin{bmatrix}
+& -3 & 0 \\
+           -4 & 1 &1
+         \end{bmatrix}</math> and <math>B=
+    \begin{bmatrix}
+& 1\\
+& 0 \\
+           -1 & 1
+         \end{bmatrix}.</math> Find <math>AB,</math> <math>BA^T</math> and <math>A-B^T.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;

Difference between revisions of "031 Review Problems"

Revision as of 17:00, 24 August 2017

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