Difference between revisions of "031 Review Problems"

Latest revision as of 13:11, 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.}

Solution:

Final Answer:

2. True or false: If a matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^2} is diagonalizable, then the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} must be diagonalizable as well.

Solution:

Final Answer:

3. True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\times 4} matrix with characteristic equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda(\lambda-1)(\lambda+1)(\lambda+e)=0,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is diagonalizable.

Solution:

Final Answer:

4. True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is invertible, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is diagonalizable.

Solution:

Final Answer:

5. True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} are invertible Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n} matrices, then so is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A+B.}

Solution:

Final Answer:

6. True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 5} matrix and Failed to parse (MathML with SVG or PNG fallback (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.}

Solution:

Final Answer:

7. True or false: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C=AB} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4\times 4} matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B.} If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is invertible, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is invertible.

Solution:

Final Answer:

8. True or false: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} be a subspace of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^4} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} be a vector in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^4.} If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\in W} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\in W^\perp,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}=\vec{0}.}

Solution:

Final Answer:

9. True or false: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is an invertible Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3} matrix, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\times 3} matrices such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB=AC,} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B=C.}

Solution:

Final Answer:

10.

(a) Is the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 3 & 1 \\ 0 & 3 \end{bmatrix}} diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

(b) Is the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 2 & 0 & -2 \\ 1 & 3 & 2 \\ 0 & 0 & 3 \end{bmatrix}} diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

Solution:

Final Answer:

11. Find the eigenvalues and eigenvectors of the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 1 & 1 & 1 \\ 0 & -1 & 1 \\ 0 & 0 & 2 \end{bmatrix}.}

Solution:

Final Answer:

12. Consider the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A= \begin{bmatrix} 1 & -4 & 9 & -7 \\ -1 & 2 & -4 & 1 \\ 5 & -6 & 10 & 7 \end{bmatrix}} and assume that it is row equivalent to the matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B= \begin{bmatrix} 1 & 0 & -1 & 5 \\ 0 & -2 & 5 & -6 \\ 0 & 0 & 0 & 0 \end{bmatrix}.}

(a) List rank Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (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 ${\text{Col }}A$ and ${\text{Nul }}A.$ Find an example of a nonzero vector that belongs to ${\text{Col }}A,$ as well as an example of a nonzero vector that belongs to ${\text{Nul }}A.$

Solution:

Final Answer:

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}}

Solution:

Final Answer:

14. Let $B={\begin{bmatrix}1&-2&3&4\\0&3&0&0\\0&5&1&2\\0&-1&3&6\end{bmatrix}}.$

(a) Is $B$ invertible? Explain.

(b) Define a linear transformation $T$ by the formula $T({\vec {x}})=B{\vec {x}}.$ Is $T$ onto? Explain.

Solution:

Final Answer:

15. Suppose $T$ is a linear transformation given by the formula

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 $T.$

(b) Let ${\vec {u}}=7{\vec {e_{1}}}-4{\vec {e_{2}}}.$ Find $T({\vec {u}}).$

(c) Is ${\begin{bmatrix}-1\\3\end{bmatrix}}$ in the range of $T?$ Explain.

Solution:

Final Answer:

16. Let $A$ and $B$ be $6\times 6$ matrices with ${\text{det }}A=-10$ and ${\text{det }}B=5.$ Use properties of determinants to compute:

(a) ${\text{det }}3A$

(b) ${\text{det }}(A^{T}B^{-1})$

Solution:

Final Answer:

17. Let $A={\begin{bmatrix}5&1\\0&5\end{bmatrix}}.$

(a) Find a basis for the eigenspace(s) of $A.$

(b) Is the matrix $A$ diagonalizable? Explain.

Solution:

Final Answer:

18. Let ${\vec {v}}={\begin{bmatrix}-1\\3\\0\end{bmatrix}}$ and ${\vec {y}}={\begin{bmatrix}2\\0\\5\end{bmatrix}}.$

(a) Find a unit vector in the direction of ${\vec {v}}.$

(b) Find the distance between ${\vec {v}}$ and ${\vec {y}}.$

(c) Let $L={\text{Span }}\{{\vec {v}}\}.$ Compute the orthogonal projection of ${\vec {y}}$ onto $L.$

Solution:

Final Answer:

19. Let $W={\text{Span }}{\Bigg \{}{\begin{bmatrix}2\\0\\-1\\0\end{bmatrix}},{\begin{bmatrix}-3\\1\\0\\0\end{bmatrix}}{\Bigg \}}.$ Is ${\begin{bmatrix}2\\6\\4\\0\end{bmatrix}}$ in $W^{\perp }?$ Explain.

Solution:

Final Answer:

20.

(a) Let $T:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}$ be a transformation given by

T{\bigg (}{\begin{bmatrix}x\\y\end{bmatrix}}{\bigg )}={\begin{bmatrix}1-xy\\x+y\end{bmatrix}}.

Determine whether $T$ is a linear transformation. Explain.

(b) Let $A={\begin{bmatrix}1&-3&0\\-4&1&1\end{bmatrix}}$ and $B={\begin{bmatrix}2&1\\1&0\\-1&1\end{bmatrix}}.$ Find $AB,~BA^{T}$ and $A-B^{T}.$

Solution:

Final Answer:

21. Let $A={\begin{bmatrix}1&3&8\\2&4&11\\1&2&5\end{bmatrix}}.$ Find $A^{-1}$ if possible.

Solution:

Final Answer:

22. Find a formula for ${\begin{bmatrix}1&-6\\2&-6\end{bmatrix}}^{k}$ by diagonalizing the matrix.

Solution:

Final Answer:

23.

(a) Show that if ${\vec {x}}$ is an eigenvector of the matrix $A$ corresponding to the eigenvalue 2, then ${\vec {x}}$ is an eigenvector of $A^{3}-A^{2}+I.$ What is the corresponding eigenvalue?

(b) Show that if ${\vec {y}}$ is an eigenvector of the matrix $A$ corresponding to the eigenvalue 3 and $A$ is invertible, then ${\vec {y}}$ is an eigenvector of $A^{-1}.$ What is the corresponding eigenvalue?

Solution:

Final Answer:

24. Let $A={\begin{bmatrix}3&0&-1\\0&1&-3\\1&0&0\end{bmatrix}}{\begin{bmatrix}3&0&0\\0&4&0\\0&0&3\end{bmatrix}}{\begin{bmatrix}0&0&1\\-3&1&9\\-1&0&3\end{bmatrix}}.$

Use the Diagonalization Theorem to find the eigenvalues of $A$ and a basis for each eigenspace.

Solution:

Final Answer:

25. Give an example of a $3\times 3$ matrix $A$ with eigenvalues 5,-1 and 3.

Solution:

Final Answer:

26. Assume $A^{2}=I.$ Find ${\text{Nul }}A.$

Solution:

Final Answer:

27. If $A$ is an $n\times n$ matrix such that $AA^{T}=I,$ what are the possible values of ${\text{det }}A?$

Solution:

Final Answer:

28. Show that if ${\vec {x}}$ is an eigenvector of the matrix product $AB$ and $B{\vec {x}}\neq {\vec {0}},$ then $B{\vec {x}}$ is an eigenvector of $BA.$

Solution:

Final Answer:

29.

(a) Suppose a $6\times 8$ matrix $A$ has 4 pivot columns. What is ${\text{dim Nul }}A?$ Is ${\text{Col }}A=\mathbb {R} ^{4}?$ Why or why not?

(b) If $A$ is a $7\times 5$ matrix, what is the smallest possible dimension of ${\text{Nul }}A?$

Solution:

Final Answer:

30. Consider the following system of equations.

x_{1}+kx_{2}=1

3x_{1}+5x_{2}=2k

Find all real values of $k$ such that the system has only one solution.

Solution:

Final Answer:

31. Suppose $\{{\vec {u}},{\vec {v}}\}$ is a basis of the eigenspace corresponding to the eigenvalue 0 of a $5\times 5$ matrix $A.$

(a) Is ${\vec {w}}={\vec {u}}-2{\vec {v}}$ an eigenvector of $A?$ If so, find the corresponding eigenvalue.

If not, explain why.

(b) Find the dimension of ${\text{Col }}A.$

Solution:

Final Answer:

@@ Line 185: / Line 185: @@
 |}
-'''12.''' Consider the matrix &nbsp;<math>A=
+'''12.''' Consider the matrix &nbsp;<math style="vertical-align: -31px">A=
      \begin{bmatrix}
 & -4 & 9 & -7 \\
@@ Line 199: / Line 199: @@
           \end{bmatrix}.</math>
-(a) List rank &nbsp;<math>A</math>&nbsp; and &nbsp;<math>\text{dim Nul }A.</math>
+(a) List rank &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{dim Nul }A.</math>
-(b) Find bases for &nbsp;<math>\text{Col }A</math>&nbsp; and &nbsp;<math>\text{Nul }A.</math>&nbsp; Find an example of a nonzero vector that belongs to &nbsp;<math>\text{Col }A,</math>&nbsp; as well as an example of a nonzero vector that belongs to &nbsp;<math>\text{Nul }A.</math>
+(b) Find bases for &nbsp;<math style="vertical-align: 0px">\text{Col }A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{Nul }A.</math>&nbsp; Find an example of a nonzero vector that belongs to &nbsp;<math style="vertical-align: -5px">\text{Col }A,</math>&nbsp; as well as an example of a nonzero vector that belongs to &nbsp;<math style="vertical-align: 0px">\text{Nul }A.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 262: / Line 262: @@
   </math>
-(a) Is &nbsp;<math>B</math>&nbsp; invertible? Explain.
+(a) Is &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; invertible? Explain.
-(b) Define a linear transformation &nbsp;<math>T</math>&nbsp; by the formula &nbsp;<math>T(\vec{x})=B\vec{x}.</math>&nbsp; Is &nbsp;<math>T</math>&nbsp; onto? Explain.
+(b) Define a linear transformation &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; by the formula &nbsp;<math style="vertical-align: -5px">T(\vec{x})=B\vec{x}.</math>&nbsp; Is &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; onto? Explain.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 279: / Line 279: @@
 |}
-'''15.''' Suppose &nbsp;<math>T</math>&nbsp; is a linear transformation given by the formula
+'''15.''' Suppose &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation given by the formula
 ::<math>T\Bigg(
@@ Line 293: / Line 293: @@
           \end{bmatrix}</math>
-(a) Find the standard matrix for &nbsp;<math>T.</math>
+(a) Find the standard matrix for &nbsp;<math style="vertical-align: 0px">T.</math>
-(b) Let &nbsp;<math>\vec{u}=7\vec{e_1}-4\vec{e_2}.</math>&nbsp; Find &nbsp;<math>T(\vec{u}).</math>
+(b) Let &nbsp;<math style="vertical-align: -5px">\vec{u}=7\vec{e_1}-4\vec{e_2}.</math>&nbsp; Find &nbsp;<math style="vertical-align: -6px">T(\vec{u}).</math>
-(c) Is &nbsp;<math>\begin{bmatrix}
+(c) Is &nbsp;<math style="vertical-align: -21px">\begin{bmatrix}
             -1 \\
-          \end{bmatrix}</math>&nbsp; in the range of &nbsp;<math>T?</math>&nbsp; Explain.
+          \end{bmatrix}</math>&nbsp; in the range of &nbsp;<math style="vertical-align: 0px">T?</math>&nbsp; Explain.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 315: / Line 315: @@
 |}
-'''16.''' Let &nbsp;<math>A</math>&nbsp; and &nbsp;<math>B</math>&nbsp; be &nbsp;<math>6\times 6</math>&nbsp; matrices with &nbsp;<math>\text{det }A=-10</math>&nbsp; and &nbsp;<math>\text{det }B=5.</math>&nbsp; Use properties of determinants to compute:
+'''16.''' Let &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; be &nbsp;<math style="vertical-align: 0px">6\times 6</math>&nbsp; matrices with &nbsp;<math style="vertical-align: -1px">\text{det }A=-10</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{det }B=5.</math>&nbsp; Use properties of determinants to compute:
-(a) &nbsp;<math>\text{det }3A</math>
+(a) &nbsp;<math style="vertical-align: -2px">\text{det }3A</math>
-(b) &nbsp;<math>\text{det }(A^TB^{-1})</math>
+(b) &nbsp;<math style="vertical-align: -7px">\text{det }(A^TB^{-1})</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 334: / Line 334: @@
 |}
-'''17.''' Let &nbsp;<math>A=
+'''17.''' Let &nbsp;<math style="vertical-align: -20px">A=
      \begin{bmatrix}
 & 1 \\
@@ Line 340: / Line 340: @@
           \end{bmatrix}.</math>
-(a) Find a basis for the eigenspace(s) of &nbsp;<math>A.</math>
+(a) Find a basis for the eigenspace(s) of &nbsp;<math style="vertical-align: 0px">A.</math>
-(b) Is the matrix &nbsp;<math>A</math>&nbsp; diagonalizable? Explain.
+(b) Is the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; diagonalizable? Explain.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 367: / Line 367: @@
           \end{bmatrix}.</math>
-(a) Find a unit vector in the direction of &nbsp;<math>\vec{v}.</math>
+(a) Find a unit vector in the direction of &nbsp;<math style="vertical-align: 0px">\vec{v}.</math>
-(b) Find the distance between &nbsp;<math>\vec{v}</math>&nbsp; and &nbsp;<math>\vec{y}.</math>
+(b) Find the distance between &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; and &nbsp;<math style="vertical-align: -3px">\vec{y}.</math>
-(c) Let &nbsp;<math>L=\text{Span }\{\vec{v}\}.</math>&nbsp; Compute the orthogonal projection of &nbsp;<math>\vec{y}</math>&nbsp; onto &nbsp;<math>L.</math>
+(c) Let &nbsp;<math style="vertical-align: -5px">L=\text{Span }\{\vec{v}\}.</math>&nbsp; Compute the orthogonal projection of &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; onto &nbsp;<math style="vertical-align: 0px">L.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 401: / Line 401: @@
 \\
-          \end{bmatrix}</math>&nbsp; in &nbsp;<math>W^\perp?</math>&nbsp; Explain.
+          \end{bmatrix}</math>&nbsp; in &nbsp;<math style="vertical-align: 0px">W^\perp?</math>&nbsp; Explain.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 418: / Line 418: @@
 '''20.'''
-(a) Let &nbsp;<math>T:\mathbb{R}^2\rightarrow \mathbb{R}^2</math>&nbsp; be a transformation given by
+(a) Let &nbsp;<math style="vertical-align: -2px">T:\mathbb{R}^2\rightarrow \mathbb{R}^2</math>&nbsp; be a transformation given by
 ::<math>T\bigg(
@@ Line 431: / Line 431: @@
           \end{bmatrix}.</math>
-Determine whether &nbsp;<math>T</math>&nbsp; is a linear transformation. Explain.
+Determine whether &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation. Explain.
-(b) Let &nbsp;<math>A=
+(b) Let &nbsp;<math style="vertical-align: -19px">A=
      \begin{bmatrix}
 & -3 & 0 \\
             -4 & 1 &1
-          \end{bmatrix}</math>&nbsp; and &nbsp;<math>B=
+          \end{bmatrix}</math>&nbsp; and &nbsp;<math style="vertical-align: -32px">B=
      \begin{bmatrix}
 & 1\\
 & 0 \\
             -1 & 1
-          \end{bmatrix}.</math>&nbsp; Find &nbsp;<math>AB,</math>&nbsp; &nbsp;<math>BA^T</math>&nbsp; and &nbsp;<math>A-B^T.</math>
+          \end{bmatrix}.</math>&nbsp; Find &nbsp;<math style="vertical-align: -4px">AB,~BA^T</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">A-B^T.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 457: / Line 457: @@
 |}
-'''21.''' Let &nbsp;<math>A=
+'''21.''' Let &nbsp;<math style="vertical-align: -31px">A=
      \begin{bmatrix}
 & 3 & 8 \\
 & 4 &11\\
 & 2 & 5
-          \end{bmatrix}.</math>&nbsp; Find &nbsp;<math>A^{-1}</math>&nbsp; if possible.
+          \end{bmatrix}.</math>&nbsp; Find &nbsp;<math style="vertical-align: 0px">A^{-1}</math>&nbsp; if possible.
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@@ Line 497: / Line 497: @@
 '''23.'''
-(a) Show that if &nbsp;<math>\vec{x}</math>&nbsp; is an eigenvector of the matrix &nbsp;<math>A</math>&nbsp; corresponding to the eigenvalue 2, then &nbsp;<math>\vec{x}</math>&nbsp; is an eigenvector of &nbsp;<math>A^3-A^2+I.</math>&nbsp; What is the corresponding eigenvalue?
+(a) Show that if &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; is an eigenvector of the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; corresponding to the eigenvalue 2, then &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; is an eigenvector of &nbsp;<math style="vertical-align: -2px">A^3-A^2+I.</math>&nbsp; What is the corresponding eigenvalue?
-(b) Show that if &nbsp;<math>\vec{y}</math>&nbsp; is an eigenvector of the matrix &nbsp;<math>A</math>&nbsp; corresponding to the eigenvalue 3 and &nbsp;<math>A</math>&nbsp; is invertible, then &nbsp;<math>\vec{y}</math>&nbsp; is an eigenvector of &nbsp;<math>A^{-1}.</math>&nbsp; What is the corresponding eigenvalue?
+(b) Show that if &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; is an eigenvector of the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; corresponding to the eigenvalue 3 and &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is invertible, then &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; is an eigenvector of &nbsp;<math style="vertical-align: 0px">A^{-1}.</math>&nbsp; What is the corresponding eigenvalue?
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@@ Line 528: / Line 528: @@
           \end{bmatrix}.</math>
-Use the Diagonalization Theorem to find the eigenvalues of &nbsp;<math>A</math>&nbsp; and a basis for each eigenspace.
+Use the Diagonalization Theorem to find the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and a basis for each eigenspace.
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@@ Line 543: / Line 543: @@
 |}
-'''25.''' Give an example of a &nbsp;<math>3\times 3</math>&nbsp; matrix &nbsp;<math>A</math>&nbsp; with eigenvalues 5,-1 and 3.
+'''25.''' Give an example of a &nbsp;<math style="vertical-align: 0px">3\times 3</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; with eigenvalues 5,-1 and 3.
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@@ Line 558: / Line 558: @@
 |}
-'''26.''' Assume &nbsp;<math>A^2=I.</math>&nbsp; Find &nbsp;<math>\text{Nul }A.</math>
+'''26.''' Assume &nbsp;<math style="vertical-align: 0px">A^2=I.</math>&nbsp; Find &nbsp;<math style="vertical-align: -1px">\text{Nul }A.</math>
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@@ Line 573: / Line 573: @@
 |}
-'''27.''' If &nbsp;<math>A</math>&nbsp; is an &nbsp;<math>n\times n</math>&nbsp; matrix such that &nbsp;<math>AA^T=I,</math>&nbsp; what are the possible values of &nbsp;<math>\text{det }A?</math>
+'''27.''' If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is an &nbsp;<math style="vertical-align: 0px">n\times n</math>&nbsp; matrix such that &nbsp;<math style="vertical-align: -4px">AA^T=I,</math>&nbsp; what are the possible values of &nbsp;<math style="vertical-align: 0px">\text{det }A?</math>
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@@ Line 588: / Line 588: @@
 |}
-'''28.''' Show that if &nbsp;<math>\vec{x}</math>&nbsp; is an eigenvector of the matrix product &nbsp;<math>AB</math>&nbsp; and &nbsp;<math>B\vec{x}\ne \vec{0},</math>&nbsp; then &nbsp;<math>B\vec{x}</math>&nbsp; is an eigenvector of &nbsp;<math>BA.</math>
+'''28.''' Show that if &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; is an eigenvector of the matrix product &nbsp;<math style="vertical-align: 0px">AB</math>&nbsp; and &nbsp;<math style="vertical-align: -5px">B\vec{x}\ne \vec{0},</math>&nbsp; then &nbsp;<math style="vertical-align: 0px">B\vec{x}</math>&nbsp; is an eigenvector of &nbsp;<math style="vertical-align: 0px">BA.</math>
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@@ Line 605: / Line 605: @@
 '''29.'''
-(a) Suppose a &nbsp;<math>6\times 8</math>&nbsp; matrix &nbsp;<math>A</math>&nbsp; has 4 pivot columns. What is &nbsp;<math>\text{dim Nul }A?</math>&nbsp; Is &nbsp;<math>\text{Col }A=\mathbb{R}^4?</math>&nbsp; Why or why not?
+(a) Suppose a &nbsp;<math style="vertical-align: 0px">6\times 8</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; has 4 pivot columns. What is &nbsp;<math style="vertical-align: -1px">\text{dim Nul }A?</math>&nbsp; Is &nbsp;<math style="vertical-align: -1px">\text{Col }A=\mathbb{R}^4?</math>&nbsp; Why or why not?
-(b) If &nbsp;<math>A</math>&nbsp; is a &nbsp;<math>7\times 5</math>&nbsp; matrix, what is the smallest possible dimension of &nbsp;<math>\text{Nul }A?</math>
+(b) If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a &nbsp;<math style="vertical-align: 0px">7\times 5</math>&nbsp; matrix, what is the smallest possible dimension of &nbsp;<math style="vertical-align: -1px">\text{Nul }A?</math>
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@@ Line 628: / Line 628: @@
 ::<math>3x_1+5x_2=2k</math>
-Find all real values of &nbsp;<math>k</math>&nbsp; such that the system has only one solution.
+Find all real values of &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; such that the system has only one solution.
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@@ Line 643: / Line 643: @@
 |}
-'''31.''' Suppose &nbsp;<math>\{\vec{u},\vec{v}\}</math>&nbsp; is a basis of the eigenspace corresponding to the eigenvalue 0 of a &nbsp;<math>5\times 5</math>&nbsp; matrix &nbsp;<math>A.</math>
+'''31.''' Suppose &nbsp;<math style="vertical-align: -5px">\{\vec{u},\vec{v}\}</math>&nbsp; is a basis of the eigenspace corresponding to the eigenvalue 0 of a &nbsp;<math style="vertical-align: 0px">5\times 5</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A.</math>
-(a) Is &nbsp;<math>\vec{w}=\vec{u}-2\vec{v}</math>&nbsp; an eigenvector of &nbsp;<math>A?</math>&nbsp; If so, find the corresponding eigenvalue.
+(a) Is &nbsp;<math style="vertical-align: 0px">\vec{w}=\vec{u}-2\vec{v}</math>&nbsp; an eigenvector of &nbsp;<math style="vertical-align: 0px">A?</math>&nbsp; If so, find the corresponding eigenvalue.
 If not, explain why.
-(b) Find the dimension of &nbsp;<math>\text{Col }A.</math>
+(b) Find the dimension of &nbsp;<math style="vertical-align: -1px">\text{Col }A.</math>
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Difference between revisions of "031 Review Problems"

Latest revision as of 13:11, 25 August 2017

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