031 Review Part 2, Problem 10 - Revision history

Kayla Murray at 20:42, 15 October 2017

2017-10-15T20:42:09Z

Kayla Murray at 20:19, 12 October 2017

2017-10-12T20:19:19Z

Kayla Murray at 20:08, 12 October 2017

2017-10-12T20:08:15Z

Kayla Murray at 16:41, 12 October 2017

2017-10-12T16:41:34Z

Kayla Murray at 04:03, 12 October 2017

2017-10-12T04:03:49Z

Kayla Murray at 02:17, 10 October 2017

2017-10-10T02:17:56Z

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

2017-10-10T01:57:58Z

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

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 <span class="exam">(b) If &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is a &nbsp;<math style="vertical-align: 0px">7\times 5</math>&nbsp; matrix, what is the smallest possible dimension of &nbsp;<math style="vertical-align: -1px">\text{Nul }A?</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 86: / Line 85: @@
 |&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math style="vertical-align: -1px">0</math>
 |}
-[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
+[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

← Older revision		Revision as of 02:17, 10 October 2017
Line 1:		Line 1:
−	<span class="exam">~~Consider the~~ matrix  <math style="vertical-align: ~~-31px~~">A=	+	<span class="exam">(a) Suppose a  <math style="vertical-align: 0px">6\times 8</math>  matrix  <math style="vertical-align: 0px">A</math>  has 4 pivot columns. What is  <math style="vertical-align: -1px">\text{dim Nul }A?</math>  Is  <math style="vertical-align: -1px">\text{Col }A=\mathbb{R}^4?</math>  Why or why not?
−	\~~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=~~	+	<span class="exam">(b) If  <math style="vertical-align: 0px">A</math>  is a  <math style="vertical-align: 0px">7\times 5</math>  matrix, what is the smallest possible dimension of  <math style="vertical-align: -1px">\text{Nul }A?</math>
−	~~\begin{bmatrix}~~
−	~~1 & 0 & -1 & 5 \\~~
−	~~0 & -2 & 5 & -6 \\~~
−	~~0 & 0 & 0 & 0~~
−	~~\end{bmatrix}.</math>~~
−
−	~~<span class="exam">(a) List rank  <math style="vertical-align: 0px">A</math>  and  <math style="vertical-align: 0px">\text{dim Nul }A.</math>~~
−
−	<span class="exam">(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>

@@ Line 12: / Line 12: @@
 |-
+|
-::<math>\text{rank }A+\text{dim Col }A=n.</math>
+::<math>\text{dim Nul }A+\text{dim Col }A=n.</math>
 |}
@@ Line 45: / Line 45: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
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 |}
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 !Step 2: &nbsp;
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@@ Line 61: / Line 84: @@
 |&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math style="vertical-align: -1px">\text{dim Col }A=4</math>&nbsp; and &nbsp;<math style="vertical-align: -6px">\text{Col }A\ne \mathbb{R}^4</math>
 |-
-|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;
+|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math style="vertical-align: -1px">0</math>
 |}
 [[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]

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 !Foundations: &nbsp;
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