031 Review Part 2, Problem 4 - Revision history

Kayla Murray at 20:27, 15 October 2017

2017-10-15T20:27:48Z

Kayla Murray at 03:25, 12 October 2017

2017-10-12T03:25:32Z

Kayla Murray at 03:12, 12 October 2017

2017-10-12T03:12:34Z

Kayla Murray at 02:56, 12 October 2017

2017-10-12T02:56:51Z

Kayla Murray at 22:18, 11 October 2017

2017-10-11T22:18:32Z

Kayla Murray at 02:09, 10 October 2017

2017-10-10T02:09:16Z

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

2017-10-10T02:01:31Z

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:  
|-
|
|}

'''Solution:'''

'''(a)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|
|}

'''(b)'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|   '''(a)'''    
|-
|   '''(b)'''    
|}
[[031_Review_Part_2|'''Return to Sample Exam''']]

@@ Line 21: / Line 21: @@
           \end{bmatrix}</math>&nbsp; in the range of &nbsp;<math style="vertical-align: 0px">T?</math>&nbsp; Explain.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 37: / Line 36: @@
 :where &nbsp;<math style="vertical-align: -5px">\{e_1,e_2,\ldots,e_n\}</math>&nbsp; is the standard basis of &nbsp;<math style="vertical-align: -1px">\mathbb{R}^n.</math>
 |-
-|'''2.''' A vector &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; is in the image of &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; if there exists &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; such that
+|'''2.''' A vector &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; is in the image of &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; if there exists &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; such that
 |-
+|
@@ Line 58: / Line 57: @@
 \\
             -1
-          \end{bmatrix},T(\vec{e_2})=
+          \end{bmatrix},~T(\vec{e_2})=
   \begin{bmatrix}
             -2.5 \\
 .5
-          \end{bmatrix},T(\vec{e_3})=
+          \end{bmatrix},\text{ and }T(\vec{e_3})=
   \begin{bmatrix}
 \\
@@ Line 78: / Line 77: @@
 & -2.5 &10 \\
             -1 & 0.5 & -2
-          \end{bmatrix}</math>
+          \end{bmatrix}.</math>
 |}
@@ Line 185: / Line 184: @@
           \end{bmatrix}</math>
 |-
-|&nbsp;&nbsp; '''(c)''' &nbsp; &nbsp; See above
+|&nbsp;&nbsp; '''(c)''' &nbsp; &nbsp; No, see above
 |}
-[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
+[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

@@ Line 132: / Line 132: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
 |-
+|
 |}
@@ Line 140: / Line 148: @@
 |-
+|
 |}
@@ Line 146: / Line 175: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;
+|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math>[T]=\begin{bmatrix}
 |-
-|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;
+|&nbsp;&nbsp; '''(c)''' &nbsp; &nbsp; See above
 |}
 [[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 85: / Line 85: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
 |-
+|
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 2: &nbsp;
 |-
+|
 |}

@@ Line 25: / Line 25: @@
 {| 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"> Suppose &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation given by the formula
-::<math>B=
+::<math>T\Bigg(
-    \begin{bmatrix}
+\begin{bmatrix}
-& 0 & -1 & 5 \\
+            x_1 \\
-& -2  & 5 & -6 \\
+           x_2 \\
-& 0 & 0 & 0
+           x_3 \\
-          \end{bmatrix}.</math>
+         \end{bmatrix}
+         \Bigg)=
-<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>
+ \begin{bmatrix}
+x_1-2.5x_2+10x_3 \\
-<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>
+            -x_1+0.5x_2-2x_3
@@ Line 42: / Line 47: @@
 '''(b)'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"