031 Review Part 3, Problem 4 - Revision history

Kayla Murray at 20:56, 15 October 2017

2017-10-15T20:56:07Z

Kayla Murray at 17:00, 13 October 2017

2017-10-13T17:00:05Z

Kayla Murray at 16:57, 13 October 2017

2017-10-13T16:57:23Z

Kayla Murray at 16:43, 13 October 2017

2017-10-13T16:43:28Z

Kayla Murray at 02:24, 10 October 2017

2017-10-10T02:24:12Z

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

2017-10-10T02:03:24Z

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:  
|-
|   '''(a)'''    
|-
|   '''(b)'''    
|}
[[031_Review_Part_3|'''Return to Sample Exam''']]

@@ Line 15: / Line 15: @@
           \end{bmatrix}</math>&nbsp; in &nbsp;<math style="vertical-align: 0px">W^\perp?</math>&nbsp; Explain.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 110: / Line 109: @@
           \end{bmatrix}\in W^\perp</math>
 |}
-[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
+[[031_Review_Part_3|'''<u>Return to Review Problems</u>''']]

@@ Line 30: / Line 30: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
 |-
+|
 |}
@@ Line 39: / Line 70: @@
 |-
+|
 |}
@@ Line 45: / Line 101: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; &nbsp; &nbsp;
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{bmatrix}
 |}
 [[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"> Let &nbsp;<math>W=\text{Span }\Bigg\{\begin{bmatrix}
-    \begin{bmatrix}
+\\
-& -4 & 9 & -7 \\
+\\
-            -1 & 2  & -4 & 1 \\
+            -1 \\
-& -6 & 10 & 7
-          \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
+          \end{bmatrix},\begin{bmatrix}
+            -3 \\
-::<math>B=
+\\
-    \begin{bmatrix}
+\\
-& 0 & -1 & 5 \\
-& -2  & 5 & -6 \\
+          \end{bmatrix}\Bigg\}.</math>&nbsp; Is &nbsp;<math>\begin{bmatrix}
-& 0 & 0 & 0
+\\
-          \end{bmatrix}.</math>
+\\
+\\
-<span class="exam">(a) List rank &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">\text{dim Nul }A.</math>
+         \end{bmatrix}</math>&nbsp; in &nbsp;<math style="vertical-align: 0px">W^\perp?</math>&nbsp; Explain.
@@ Line 26: / Line 25: @@
 '''Solution:'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 59: / Line 42: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;
+|&nbsp;&nbsp; &nbsp; &nbsp;
 |}
 [[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 20: / Line 20: @@
 !Foundations: &nbsp;
 |-
-|Recall that if &nbsp;<math>W</math>&nbsp; is a subspace of &nbsp;<math>\mathbb{R}^n,</math>&nbsp; then
+|Recall that if &nbsp;<math style="vertical-align: 0px">W</math>&nbsp; is a subspace of &nbsp;<math style="vertical-align: -4px">\mathbb{R}^n,</math>&nbsp; then
 |-
+|
@@ Line 42: / Line 42: @@
           \end{bmatrix}</math>
 |-
-|is in  &nbsp;<math>W^\perp,</math>&nbsp; it suffices to see if this vector is orthogonal to
+|is in  &nbsp;<math style="vertical-align: -4px">W^\perp,</math>&nbsp; it suffices to see if this vector is orthogonal to
 |-
-|the basis elements of &nbsp;<math>W.</math>
+|the basis elements of &nbsp;<math style="vertical-align: 0px">W.</math>
 |-
 |Notice that we have
@@ Line 68: / Line 68: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 2: &nbsp;
 |-
+|