031 Review Part 2 - Revision history

Kayla Murray at 02:34, 10 October 2017

2017-10-10T02:34:29Z

Kayla Murray at 02:19, 10 October 2017

2017-10-10T02:19:31Z

Kayla Murray: /* Problem 7 */

2017-10-10T01:57:26Z

Problem 7

Kayla Murray at 01:48, 10 October 2017

2017-10-10T01:48:09Z

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
2017-10-09T18:21:25Z

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 ]]==

Consider the matrix  <math style="vertical-align: -31px">A=
\begin{bmatrix}
1 & -4 & 9 & -7 \\
-1 & 2 & -4 & 1 \\
5 & -6 & 10 & 7
\end{bmatrix}</math>  and assume that it is row equivalent to the matrix

::<math>B=
\begin{bmatrix}
1 & 0 & -1 & 5 \\
0 & -2 & 5 & -6 \\
0 & 0 & 0 & 0
\end{bmatrix}.</math>

(a) List rank  <math style="vertical-align: 0px">A</math>  and  <math style="vertical-align: 0px">\text{dim Nul }A.</math>

(b) Find bases for  <math style="vertical-align: 0px">\text{Col }A</math>  and  <math style="vertical-align: 0px">\text{Nul }A.</math>  Find an example of a nonzero vector that belongs to  <math style="vertical-align: -5px">\text{Col }A,</math>  as well as an example of a nonzero vector that belongs to  <math style="vertical-align: 0px">\text{Nul }A.</math>

== [[009C_Sample Final 1,_Problem_2| Problem 2 ]] ==
 Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?

::<math>\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}</math>

== [[009C_Sample Final 1,_Problem_3| Problem 3 ]] ==
Let
 <math>B=
\begin{bmatrix}
1 & -2 & 3 & 4\\
0 & 3 &0 &0\\
0 & 5 & 1 & 2\\
0 & -1 & 3 & 6
\end{bmatrix}.
</math>

(a) Is  <math style="vertical-align: 0px">B</math>  invertible? Explain.

(b) Define a linear transformation  <math style="vertical-align: 0px">T</math>  by the formula  <math style="vertical-align: -5px">T(\vec{x})=B\vec{x}.</math>  Is  <math style="vertical-align: 0px">T</math>  onto? Explain.

== [[009C_Sample Final 1,_Problem_4| Problem 4 ]] ==
 Suppose  <math style="vertical-align: 0px">T</math>  is a linear transformation given by the formula

::<math>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}</math>

(a) Find the standard matrix for  <math style="vertical-align: 0px">T.</math>

(b) Let  <math style="vertical-align: -5px">\vec{u}=7\vec{e_1}-4\vec{e_2}.</math>  Find  <math style="vertical-align: -6px">T(\vec{u}).</math>

(c) Is  <math style="vertical-align: -21px">\begin{bmatrix}
-1 \\
3
\end{bmatrix}</math>  in the range of  <math style="vertical-align: 0px">T?</math>  Explain.

== [[009C_Sample Final 1,_Problem_5| Problem 5 ]] ==
Let  <math style="vertical-align: 0px">A</math>  and  <math style="vertical-align: 0px">B</math>  be  <math style="vertical-align: 0px">6\times 6</math>  matrices with  <math style="vertical-align: -1px">\text{det }A=-10</math>  and  <math style="vertical-align: 0px">\text{det }B=5.</math>  Use properties of determinants to compute:

(a)  <math style="vertical-align: -2px">\text{det }3A</math>

(b)  <math style="vertical-align: -7px">\text{det }(A^TB^{-1})</math>

== [[009C_Sample Final 1,_Problem_6| Problem 6 ]] ==
 Let  <math>\vec{v}=\begin{bmatrix}
-1 \\
3 \\
0
\end{bmatrix}</math>  and  <math>\vec{y}=\begin{bmatrix}
2 \\
0 \\
5
\end{bmatrix}.</math>

(a) Find a unit vector in the direction of  <math style="vertical-align: 0px">\vec{v}.</math>

(b) Find the distance between  <math style="vertical-align: 0px">\vec{v}</math>  and  <math style="vertical-align: -3px">\vec{y}.</math>

(c) Let  <math style="vertical-align: -5px">L=\text{Span }\{\vec{v}\}.</math>  Compute the orthogonal projection of  <math style="vertical-align: -3px">\vec{y}</math>  onto  <math style="vertical-align: 0px">L.</math>

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

(a) Let  <math style="vertical-align: -2px">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}
1-xy \\
x+y
\end{bmatrix}.</math>

Determine whether  <math style="vertical-align: 0px">T</math>  is a linear transformation. Explain.

(b) Let  <math style="vertical-align: -19px">A=
\begin{bmatrix}
1 & -3 & 0 \\
-4 & 1 &1
\end{bmatrix}</math>  and  <math style="vertical-align: -32px">B=
\begin{bmatrix}
2 & 1\\
1 & 0 \\
-1 & 1
\end{bmatrix}.</math>  Find  <math style="vertical-align: -4px">AB,~BA^T</math>  and  <math style="vertical-align: 0px">A-B^T.</math>

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

Let  <math style="vertical-align: -31px">A=
\begin{bmatrix}
1 & 3 & 8 \\
2 & 4 &11\\
1 & 2 & 5
\end{bmatrix}.</math>  Find  <math style="vertical-align: 0px">A^{-1}</math>  if possible.

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

If  <math style="vertical-align: 0px">A</math>  is an  <math style="vertical-align: 0px">n\times n</math>  matrix such that  <math style="vertical-align: -4px">AA^T=I,</math>  what are the possible values of  <math style="vertical-align: 0px">\text{det }A?</math>

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

(a) Suppose a  <math style="vertical-align: 0px">6\times 8</math>  matrix  <math style="vertical-align: 0px">A</math>  has 4 pivot columns. What is  <math style="vertical-align: -1px">\text{dim Nul }A?</math>  Is  <math style="vertical-align: -1px">\text{Col }A=\mathbb{R}^4?</math>  Why or why not?

(b) If  <math style="vertical-align: 0px">A</math>  is a  <math style="vertical-align: 0px">7\times 5</math>  matrix, what is the smallest possible dimension of  <math style="vertical-align: -1px">\text{Nul }A?</math>

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

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 style="vertical-align: 0px">k</math>  such that the system has only one solution.

← 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 23: / Line 23: @@
 <span class="exam">(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>
 == [[031_Review Part 2,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
@@ Line 48: / Line 47: @@
           \end{bmatrix}</math>
 == [[031_Review Part 2,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
@@ Line 88: / Line 86: @@
           \end{bmatrix}</math>&nbsp; in the range of &nbsp;<math style="vertical-align: 0px">T?</math>&nbsp; Explain.
 == [[031_Review Part 2,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
@@ Line 96: / Line 93: @@
 <span class="exam">(b) &nbsp;<math style="vertical-align: -7px">\text{det }(A^TB^{-1})</math>
 == [[031_Review Part 2,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
@@ Line 155: / Line 151: @@
 <span class="exam">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>
 == [[031_Review Part 2,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==

@@ Line 132: / Line 132: @@
 <span class="exam">Determine whether &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation. Explain.
-<span class+"exam">(b) Let &nbsp;<math style="vertical-align: -19px">A=
+<span class="exam">(b) Let &nbsp;<math style="vertical-align: -19px">A=
      \begin{bmatrix}
 & -3 & 0 \\

@@ 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 2,_Problem_1|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 1&nbsp;</span></span>]]==
 <span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=
@@ Line 25: / Line 25: @@
-== [[009C_Sample Final 1,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_2|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 2&nbsp;</span>]] ==
 <span class="exam"> Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?
@@ Line 50: / Line 50: @@
-== [[009C_Sample Final 1,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_3|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 3&nbsp;</span>]] ==
 <span class="exam">Let
 &nbsp;<math>B=
@@ Line 65: / Line 65: @@
 <span class="exam">(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.
-== [[009C_Sample Final 1,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_4|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 4&nbsp;</span>]] ==
 <span class="exam"> Suppose &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation given by the formula
@@ Line 90: / Line 90: @@
-== [[009C_Sample Final 1,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_5|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 5&nbsp;</span>]] ==
 <span class="exam">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:
@@ Line 98: / Line 98: @@
-== [[009C_Sample Final 1,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_6|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 6&nbsp;</span>]] ==
 <span class="exam"> Let &nbsp;<math>\vec{v}=\begin{bmatrix}
             -1 \\
@@ Line 115: / Line 115: @@
 <span class="exam">(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>
-== [[009C_Sample Final 1,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_7|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 7&nbsp;</span>]] ==
 <span class="exam">(a) Let &nbsp;<math style="vertical-align: -2px">T:\mathbb{R}^2\rightarrow \mathbb{R}^2</math>&nbsp; be a transformation given by
@@ Line 143: / Line 143: @@
           \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>
-== [[009C_Sample Final 1,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_8|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 8&nbsp;</span>]] ==
 <span class="exam">Let &nbsp;<math style="vertical-align: -31px">A=
@@ Line 152: / Line 152: @@
           \end{bmatrix}.</math>&nbsp; Find &nbsp;<math style="vertical-align: 0px">A^{-1}</math>&nbsp; if possible.
-== [[009C_Sample Final 1,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_9|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 9&nbsp;</span>]] ==
 <span class="exam">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>
-== [[009C_Sample Final 1,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_10|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 10&nbsp;</span>]] ==
 <span class="exam">(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?
@@ Line 163: / Line 163: @@
 <span class="exam">(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>
-== [[009C_Sample Final 1,_Problem_11|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 11&nbsp;</span>]] ==
+== [[031_Review Part 2,_Problem_11|<span class="biglink"><span style="font-size:80%">&nbsp;Problem 11&nbsp;</span>]] ==
 <span class="exam">Consider the following system of equations.