031 Review Part 3, Problem 11 - Revision history

Kayla Murray at 21:14, 15 October 2017

2017-10-15T21:14:00Z

Kayla Murray at 17:11, 11 October 2017

2017-10-11T17:11:44Z

Kayla Murray at 04:10, 11 October 2017

2017-10-11T04:10:17Z

Kayla Murray at 02:32, 10 October 2017

2017-10-10T02:32:46Z

Kayla Murray: Created page with "Consider the matrix $A= \begin{bmatrix} 1 & -4 & 9 & -7 \\ -1 & 2 & -4 & 1 \\..."$

2017-10-10T02:04:53Z

Created page with "Consider the matrix <math style="vertical-align: -31px">A= \begin{bmatrix} 1 & -4 & 9 & -7 \\ -1 & 2 & -4 & 1 \\..."

New page

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>

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'''Solution:'''

'''(a)'''

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'''(b)'''

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|   '''(a)'''    
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[[031_Review_Part_3|'''Return to Sample Exam''']]

@@ Line 6: / Line 6: @@
 <span class="exam">(b) Find the dimension of &nbsp;<math style="vertical-align: -1px">\text{Col }A.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 40: / Line 39: @@
 |-
+|
-::<math>A\vec{u}=\vec{0}</math>&nbsp; and &nbsp;<math>A\vec{v}=\vec{0}</math>
+::<math style="vertical-align: -1px">A\vec{u}=\vec{0}</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">A\vec{v}=\vec{0}</math>
 |-
 |since &nbsp;<math style="vertical-align: 0px">\vec{u}</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; are eigenvectors of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; corresponding to the eigenvalue 0.
@@ Line 56: / Line 55: @@
 & = & \displaystyle{A\vec{u}-2A\vec{v}}\\
 &&\\
-& = & \displaystyle{\vec{0}-2\vec{0}}\\
+& = & \displaystyle{\vec{0}-2\cdot \vec{0}}\\
 &&\\
 & = & \displaystyle{\vec{0}.}
@@ Line 98: / Line 97: @@
 |&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; See solution above.
 |-
-|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math style="vertical-align: -3px">\text{dim Col }A=3.</math>
+|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math style="vertical-align: -3px">\text{dim Col }A=3</math>
 |}
-[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
+[[031_Review_Part_3|'''<u>Return to Review Problems</u>''']]

@@ Line 1: / Line 1: @@
-<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=
+<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>
-::<math>B=
+<span class="exam">(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.
-<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>
+<span class="exam">If not, explain why.