031 Review Part 2, Problem 2 - Revision history

Kayla Murray at 20:13, 15 October 2017

2017-10-15T20:13:59Z

Kayla Murray at 14:47, 11 October 2017

2017-10-11T14:47:05Z

Kayla Murray at 04:26, 11 October 2017

2017-10-11T04:26:57Z

Kayla Murray at 02:07, 10 October 2017

2017-10-10T02:07:16Z

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

2017-10-10T01:59:00Z

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:'''

{| 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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[[031_Review_Part_2|'''Return to Sample Exam''']]

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           \end{bmatrix}</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 92: / Line 91: @@
 |&nbsp;&nbsp; &nbsp; &nbsp;  The dimension is &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; and the vectors are not linearly independent.
 |}
-[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
+[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

@@ Line 39: / Line 39: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
 |-
+|
 |}
@@ Line 46: / Line 75: @@
 !Step 2: &nbsp;
 |-
+|Now, we have 3 pivots in this matrix. So, the dimension of the column space of the matrix we started with is 3.
 |}
@@ Line 53: / Line 90: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; &nbsp; &nbsp;
+|&nbsp;&nbsp; &nbsp; &nbsp;  The dimension is &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; and the vectors are not linearly independent.
 |}
 [[031_Review_Part_2|'''<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 dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?
-::<math>B=
+::<math>\begin{bmatrix}
-    \begin{bmatrix}
+\\
-& 0 & -1 & 5 \\
+\\
-& -2  & 5 & -6 \\
-& 0 & 0 & 0
+          \end{bmatrix},
-          \end{bmatrix}.</math>
+         \begin{bmatrix}
+  \\
-<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>
+\\
-−
-<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>
+         \end{bmatrix},