031 Review Part 2, Problem 7 - Revision history

Kayla Murray at 20:36, 15 October 2017

2017-10-15T20:36:44Z

Kayla Murray at 21:50, 12 October 2017

2017-10-12T21:50:09Z

Kayla Murray at 20:30, 12 October 2017

2017-10-12T20:30:13Z

Kayla Murray at 04:00, 12 October 2017

2017-10-12T04:00:27Z

Kayla Murray at 02:16, 10 October 2017

2017-10-10T02:16:08Z

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

2017-10-10T01:57:40Z

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)'''

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'''(b)'''

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|   '''(a)'''    
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|   '''(b)'''    
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[[031_Review_Part_2|'''Return to Sample Exam''']]

@@ Line 24: / Line 24: @@
             -1 & 1
           \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>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 205: / Line 204: @@
           \end{bmatrix}</math>
 |}
-[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
+[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

@@ Line 52: / Line 52: @@
 !Step 1: &nbsp;
 |-
-|We claim that &nbsp;<math>T</math>&nbsp; is not a linear transformation.
+|We claim that &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is not a linear transformation.
 |-
-|Consider the vectors &nbsp;<math>\begin{bmatrix}
+|Consider the vectors &nbsp;<math style="vertical-align: -20px">\begin{bmatrix}
 \\
-          \end{bmatrix}</math>&nbsp; and &nbsp;<math>\begin{bmatrix}
+          \end{bmatrix}</math>&nbsp; and &nbsp;<math style="vertical-align: -20px">\begin{bmatrix}
 \\
@@ Line 94: / Line 94: @@
 \\
-          \end{bmatrix}+\begin{bmatrix}
+          \end{bmatrix}\bigg)+T\bigg(\begin{bmatrix}
 \\
-          \end{bmatrix}\bigg)} & = & \displaystyle{T\bigg(\begin{bmatrix}
+          \end{bmatrix}\bigg)} & = & \displaystyle{\begin{bmatrix}
 \\
-          \end{bmatrix}\bigg)}\\
+          \end{bmatrix}}\\
 &&\\
 & = & \displaystyle{\begin{bmatrix}
 \\
           \end{bmatrix}.}
 \end{array}</math>
 |}
@@ Line 113: / Line 135: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
 |-
+|
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 2: &nbsp;
 |-
+|
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@@ Line 127: / Line 195: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;
+|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math style="vertical-align: 0px">T</math>&nbsp; is not a linear transformation
 |-
-|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;
+|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math style="vertical-align: -20px">AB=\begin{bmatrix}
 |}
 [[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]

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 !Step 1: &nbsp;
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+|
 |}
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 !Step 2: &nbsp;
 |-
+|
 |}

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 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Foundations: &nbsp;
 |-
+|
 |}

@@ Line 1: / Line 1: @@
-<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=
+<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
      \begin{bmatrix}
-& -4 & 9 & -7 \\
+& -3 & 0 \\
-            -1 & 2  & -4 & 1 \\
+            -4 & 1 &1
-& -6 & 10 & 7
+          \end{bmatrix}</math>&nbsp; and &nbsp;<math style="vertical-align: -32px">B=
      \begin{bmatrix}
-& 0 & -1 & 5 \\
+& 1\\
-& -2  & 5 & -6 \\
+& 0 \\
-& 0 & 0 & 0
+            -1 & 1
-          \end{bmatrix}.</math>
+          \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>