Difference between revisions of "031 Review Problems"

Revision as of 18:13, 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 $7\times 7$ matrix $A$ are $7,$ then det $A$ must be $7^{7}.$

Solution:

Final Answer:

2. True or false: If a matrix $A^{2}$ is diagonalizable, then the matrix $A$ must be diagonalizable as well.

Solution:

Final Answer:

3. True or false: If $A$ is a $4\times 4$ 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.

Solution:

Final Answer:

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

Solution:

Final Answer:

5. True or false: If $A$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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.}

Solution:

Final Answer:

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 $\mathbb {R} ^{3}.$

Solution:

Final Answer:

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

Solution:

Final Answer:

8. True or false: Let $W$ be a subspace of $\mathbb {R} ^{4}$ and ${\vec {v}}$ be a vector in $\mathbb {R} ^{4}.$ If ${\vec {v}}\in W$ 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}}.}

Solution:

Final Answer:

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

Solution:

Final Answer:

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 $A={\begin{bmatrix}2&0&-2\\1&3&2\\0&0&3\end{bmatrix}}$ 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 $A={\begin{bmatrix}1&1&1\\0&-1&1\\0&0&2\end{bmatrix}}.$

Solution:

Final Answer:

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

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

(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 $A.$

Solution:

Final Answer:

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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}}}

Solution:

Final Answer:

14. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B} invertible? Explain.

(b) Define a linear transformation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T} by the formula Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T({\vec {x}})=B{\vec {x}}.} Is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T} onto? Explain.

Solution:

Final Answer:

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 $T.$

(b) Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {u}}=7{\vec {e_{1}}}-4{\vec {e_{2}}}.} Find Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T({\vec {u}}).}

(c) Is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}-1\\3\end{bmatrix}}} in the range of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T?} Explain.

Solution:

Final Answer:

16. Let $A$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle B} be Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 6\times 6} matrices with det Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A=-10} and det $B=5.$ Use properties of

determinants to compute:

(a) det Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3A}

(b) det Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {v}}={\begin{bmatrix}-1\\3\\0\end{bmatrix}}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {y}}={\begin{bmatrix}2\\0\\5\end{bmatrix}}.}

(a) Find a unit vector in the direction of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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.}

Solution:

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.

Solution:

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. 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.

Solution:

Final Answer:

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.

Solution:

Final Answer:

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?

Solution:

Final Answer:

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.

Solution:

Final Answer:

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.

Solution:

Final Answer:

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 Nul Failed to parse (MathML with SVG or PNG fallback (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.}

Solution:

Final Answer:

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 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?}

Solution:

Final Answer:

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.}

Solution:

Final Answer:

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 dim Nul Failed to parse (MathML with SVG or PNG fallback (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 Col Failed to parse (MathML with SVG or PNG fallback (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=\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 Nul Failed to parse (MathML with SVG or PNG fallback (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?}

Solution:

Final Answer:

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.

Solution:

Final Answer:

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 Col Failed to parse (MathML with SVG or PNG fallback (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.}

Solution:

Final Answer:

@@ Line 459: / Line 459: @@
 |}
-'''21.'''
+'''21.''' Let <math>A=
+    \begin{bmatrix}
+& 3 & 8 \\
+& 4 &11\\
+& 2 & 5
+         \end{bmatrix}.</math> Find <math>A^{-1}</math> if possible.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 473: / Line 479: @@
 |}
-'''22.'''
+'''22.''' Find a formula for <math>\begin{bmatrix}
+& -6  \\
+& -6
+         \end{bmatrix}^k</math> by diagonalizing the matrix.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 488: / Line 498: @@
 '''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?
+(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?
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 501: / Line 516: @@
 |}
-'''24.'''
+'''24.''' Let <math>A=\begin{bmatrix}
+& 0 & -1 \\
+& 1 &-3\\
+& 0 & 0
+         \end{bmatrix}\begin{bmatrix}
+& 0 & 0 \\
+& 4 &0\\
+& 0 & 3
+         \end{bmatrix}\begin{bmatrix}
+& 0 & 1 \\
+           -3 & 1 &9\\
+           -1 & 0 & 3
+         \end{bmatrix}.</math>
+Use the Diagonalization Theorem to find the eigenvalues of <math>A</math> and a basis for each eigenspace.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 515: / Line 545: @@
 |}
-'''25.'''
+'''25.''' Give an example of a <math>3\times 3</math> matrix <math>A</math> with eigenvalues 5,-1 and 3.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 529: / Line 560: @@
 |}
-'''26.'''
+'''26.''' Assume <math>A^2=I.</math> Find Nul <math>A.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 543: / Line 575: @@
 |}
-'''27.'''
+'''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 det <math>A?</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 557: / Line 590: @@
 |}
-'''28.'''
+'''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>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
@@ Line 572: / Line 606: @@
 '''29.'''
+(a) Suppose a <math>6\times 8</math> matrix <math>A</math> has 4 pivot columns. What is dim Nul <math>A?</math> Is Col <math>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 Nul <math>A?</math>
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-'''30.'''
+'''30.''' Consider the following system of equations.
+<math>x_1+kx_2=1</math>
+<math>3x_1+5x_2=2k</math>
+Find all real values of <math>k</math> such that the system has only one solution.
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-'''31.'''
+'''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.
+If not, explain why.
+(b) Find the dimension of Col <math>A.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;

Difference between revisions of "031 Review Problems"

Revision as of 18:13, 24 August 2017

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