031 Review Part 3, Problem 2 - Revision history

Kayla Murray at 20:53, 15 October 2017

2017-10-15T20:53:19Z

Kayla Murray at 00:19, 14 October 2017

2017-10-14T00:19:10Z

Kayla Murray at 22:31, 13 October 2017

2017-10-13T22:31:50Z

Kayla Murray at 20:17, 13 October 2017

2017-10-13T20:17:14Z

Kayla Murray at 02:22, 10 October 2017

2017-10-10T02:22:42Z

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

2017-10-10T02:03:00Z

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>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
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'''Solution:'''

'''(a)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
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'''(b)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
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|   '''(a)'''    
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|   '''(b)'''    
|}
[[031_Review_Part_3|'''Return to Sample Exam''']]

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 & 0 & 2
           \end{bmatrix}.</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 70: / Line 69: @@
           \end{bmatrix}.</math>
 |}
-[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
+[[031_Review_Part_3|'''<u>Return to Review Problems</u>''']]

@@ Line 28: / Line 28: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 2: &nbsp;
 |-
+|
 |}
@@ Line 36: / Line 53: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; &nbsp; &nbsp;
+|&nbsp;&nbsp; &nbsp; &nbsp; The eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are &nbsp;<math style="vertical-align: -4px">1,-1</math>&nbsp; and &nbsp;<math style="vertical-align: -4px">2,</math>&nbsp; and the corresponding eigenvectors are
 |}
 [[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
-<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=
+<span class="exam"> Find the eigenvalues and eigenvectors of the matrix &nbsp;<math style="vertical-align: -31px">A=
      \begin{bmatrix}
-& -4 & 9 & -7 \\
+& 1 & 1 \\
-           -1 & 2  & -4 & 1 \\
+& -1  & 1 \\
-& -6 & 10 & 7
+& 0 & 2
-         \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
+          \end{bmatrix}.</math>
@@ Line 26: / Line 15: @@
 '''Solution:'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 59: / Line 32: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;
+|&nbsp;&nbsp; &nbsp; &nbsp;
 |}
 [[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 21: / Line 21: @@
 !Step 1: &nbsp;
 |-
+|Since &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a triangular matrix, the eigenvalues of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; are the entries on the diagonal.
 |}

@@ Line 10: / Line 10: @@
 !Foundations: &nbsp;
 |-
+|An eigenvector of a matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a nonzero vector &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; such that &nbsp;<math style="vertical-align: 0px">A\vec{x}=\lambda\vec{x}</math>&nbsp; for some scalar &nbsp;<math style="vertical-align: 0px">\lambda.</math>
 |}