031 Review Part 3 - Revision history

Kayla Murray at 02:34, 10 October 2017

2017-10-10T02:34:40Z

Kayla Murray at 02:32, 10 October 2017

2017-10-10T02:32:26Z

Kayla Murray at 01:50, 10 October 2017

2017-10-10T01:50:52Z

Kayla Murray: Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.''' '''Click on the
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Created page with "'''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.''' '''Click on the <span class..."

New page
'''This is a sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.'''

'''Click on the  boxed problem numbers  to go to a solution.'''
<div class="noautonum">TOC</div>

== [[009C_Sample Final 1,_Problem_1| Problem 1 ]]==

(a) Is the matrix  <math style="vertical-align: -18px">A=
\begin{bmatrix}
3 & 1 \\
0 & 3
\end{bmatrix}</math>  diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

(b) Is the matrix  <math style="vertical-align: -31px">A=
\begin{bmatrix}
2 & 0 & -2 \\
1 & 3 & 2 \\
0 & 0 & 3
\end{bmatrix}</math>  diagonalizable? If so, explain why and diagonalize it. If not, explain why not.

== [[009C_Sample Final 1,_Problem_2| Problem 2 ]] ==
 Find the eigenvalues and eigenvectors of the matrix  <math style="vertical-align: -31px">A=
\begin{bmatrix}
1 & 1 & 1 \\
0 & -1 & 1 \\
0 & 0 & 2
\end{bmatrix}.</math>

== [[009C_Sample Final 1,_Problem_3| Problem 3 ]] ==
Let  <math style="vertical-align: -20px">A=
\begin{bmatrix}
5 & 1 \\
0 & 5
\end{bmatrix}.</math>

(a) Find a basis for the eigenspace(s) of  <math style="vertical-align: 0px">A.</math>

(b) Is the matrix  <math style="vertical-align: 0px">A</math>  diagonalizable? Explain.

== [[009C_Sample Final 1,_Problem_4| Problem 4 ]] ==
 Let  <math>W=\text{Span }\Bigg\{\begin{bmatrix}
2 \\
0 \\
-1 \\
0
\end{bmatrix},\begin{bmatrix}
-3 \\
1 \\
0 \\
0
\end{bmatrix}\Bigg\}.</math>  Is  <math>\begin{bmatrix}
2 \\
6 \\
4 \\
0
\end{bmatrix}</math>  in  <math style="vertical-align: 0px">W^\perp?</math>  Explain.

== [[009C_Sample Final 1,_Problem_5| Problem 5 ]] ==
Find a formula for  <math>\begin{bmatrix}
1 & -6 \\
2 & -6
\end{bmatrix}^k</math>  by diagonalizing the matrix.

== [[009C_Sample Final 1,_Problem_6| Problem 6 ]] ==
 (a) Show that if  <math style="vertical-align: 0px">\vec{x}</math>  is an eigenvector of the matrix  <math style="vertical-align: 0px">A</math>  corresponding to the eigenvalue 2, then  <math style="vertical-align: 0px">\vec{x}</math>  is an eigenvector of  <math style="vertical-align: -2px">A^3-A^2+I.</math>  What is the corresponding eigenvalue?

(b) Show that if  <math style="vertical-align: -3px">\vec{y}</math>  is an eigenvector of the matrix  <math style="vertical-align: 0px">A</math>  corresponding to the eigenvalue 3 and  <math style="vertical-align: 0px">A</math>  is invertible, then  <math style="vertical-align: -3px">\vec{y}</math>  is an eigenvector of  <math style="vertical-align: 0px">A^{-1}.</math>  What is the corresponding eigenvalue?

== [[009C_Sample Final 1,_Problem_7| Problem 7 ]] ==

Let  <math>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}.</math>

Use the Diagonalization Theorem to find the eigenvalues of  <math style="vertical-align: 0px">A</math>  and a basis for each eigenspace.

== [[009C_Sample Final 1,_Problem_8| Problem 8 ]] ==

Give an example of a  <math style="vertical-align: 0px">3\times 3</math>  matrix  <math style="vertical-align: 0px">A</math>  with eigenvalues 5,-1 and 3.

== [[009C_Sample Final 1,_Problem_9| Problem 9 ]] ==

Assume  <math style="vertical-align: 0px">A^2=I.</math>  Find  <math style="vertical-align: -1px">\text{Nul }A.</math>

== [[009C_Sample Final 1,_Problem_10| Problem 10 ]] ==

Show that if  <math style="vertical-align: 0px">\vec{x}</math>  is an eigenvector of the matrix product  <math style="vertical-align: 0px">AB</math>  and  <math style="vertical-align: -5px">B\vec{x}\ne \vec{0},</math>  then  <math style="vertical-align: 0px">B\vec{x}</math>  is an eigenvector of  <math style="vertical-align: 0px">BA.</math>

== [[009C_Sample Final 1,_Problem_11| Problem 11 ]] ==

Suppose  <math style="vertical-align: -5px">\{\vec{u},\vec{v}\}</math>  is a basis of the eigenspace corresponding to the eigenvalue 0 of a  <math style="vertical-align: 0px">5\times 5</math>  matrix  <math style="vertical-align: 0px">A.</math>

(a) Is  <math style="vertical-align: 0px">\vec{w}=\vec{u}-2\vec{v}</math>  an eigenvector of  <math style="vertical-align: 0px">A?</math>  If so, find the corresponding eigenvalue.

If not, explain why.

(b) Find the dimension of  <math style="vertical-align: -1px">\text{Col }A.</math>

← Older revision		Revision as of 02:34, 10 October 2017
Line 1:		Line 1:
−	'''~~This is a~~ sample, and is meant to represent the material usually covered in Math 9C for the final. An actual test may or may not be similar.'''	+	'''These questions are from sample exams and actual exams at other universities. The questions are meant to represent the material usually covered in Math 31 for the final. An actual test may or may not be similar.'''

	'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''		'''Click on the <span class="biglink" style="color:darkblue;"> boxed problem numbers </span> to go to a solution.'''

@@ Line 18: / Line 18: @@
 & 0 & 3
           \end{bmatrix}</math>&nbsp; diagonalizable? If so, explain why and diagonalize it. If not, explain why not.
 == [[031_Review Part 3,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
@@ Line 28: / Line 26: @@
 & 0 & 2
           \end{bmatrix}.</math>
 == [[031_Review Part 3,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
@@ Line 58: / Line 55: @@
           \end{bmatrix}</math>&nbsp; in &nbsp;<math style="vertical-align: 0px">W^\perp?</math>&nbsp; Explain.
 == [[031_Review Part 3,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
@@ Line 96: / Line 92: @@
 <span class="exam">Assume &nbsp;<math style="vertical-align: 0px">A^2=I.</math>&nbsp; Find &nbsp;<math style="vertical-align: -1px">\text{Nul }A.</math>
 == [[031_Review Part 3,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==

@@ Line 4: / Line 4: @@
 <div class="noautonum">__TOC__</div>
-== [[009C_Sample Final 1,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]]==
+== [[031_Review Part 3,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]]==
 <span class="exam">(a) Is the matrix &nbsp;<math style="vertical-align: -18px">A=
@@ Line 21: / Line 21: @@
-== [[009C_Sample Final 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
 <span class="exam"> Find the eigenvalues and eigenvectors of the matrix &nbsp;<math style="vertical-align: -31px">A=
      \begin{bmatrix}
@@ Line 30: / Line 30: @@
-== [[009C_Sample Final 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
 <span class="exam">Let &nbsp;<math style="vertical-align: -20px">A=
      \begin{bmatrix}
@@ Line 41: / Line 41: @@
 <span class="exam">(b) Is the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; diagonalizable? Explain.
-== [[009C_Sample Final 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
 <span class="exam"> Let &nbsp;<math>W=\text{Span }\Bigg\{\begin{bmatrix}
 \\
@@ Line 60: / Line 60: @@
-== [[009C_Sample Final 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
 <span class="exam">Find a formula for &nbsp;<math>\begin{bmatrix}
 & -6  \\
@@ Line 66: / Line 66: @@
           \end{bmatrix}^k</math>&nbsp; by diagonalizing the matrix.
-== [[009C_Sample Final 1,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
 <span class="exam"> (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?
 <span class="exam">(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?
-== [[009C_Sample Final 1,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
 <span class="exam">Let &nbsp;<math>A=\begin{bmatrix}
@@ Line 89: / Line 89: @@
 <span class="exam">Use the Diagonalization Theorem to find the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and a basis for each eigenspace.
-== [[009C_Sample Final 1,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
 <span class="exam">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.
-== [[009C_Sample Final 1,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
 <span class="exam">Assume &nbsp;<math style="vertical-align: 0px">A^2=I.</math>&nbsp; Find &nbsp;<math style="vertical-align: -1px">\text{Nul }A.</math>
-== [[009C_Sample Final 1,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
 <span class="exam">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>
-== [[009C_Sample Final 1,_Problem_11|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 11&nbsp;</span>]] ==
+== [[031_Review Part 3,_Problem_11|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 11&nbsp;</span>]] ==
 <span class="exam">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>