031 Review Part 2, Problem 11 - Revision history

Kayla Murray at 20:43, 15 October 2017

2017-10-15T20:43:37Z

Kayla Murray at 00:08, 11 October 2017

2017-10-11T00:08:33Z

Kayla Murray at 18:05, 10 October 2017

2017-10-10T18:05:55Z

Kayla Murray at 17:59, 10 October 2017

2017-10-10T17:59:33Z

Kayla Murray at 17:53, 10 October 2017

2017-10-10T17:53:46Z

Kayla Murray at 17:49, 10 October 2017

2017-10-10T17:49:35Z

Kayla Murray at 02:18, 10 October 2017

2017-10-10T02:18:33Z

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

2017-10-10T02:01:17Z

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''']]

@@ Line 6: / Line 6: @@
 <span class="exam">Find all real values of &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; such that the system has only one solution.
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
@@ Line 68: / Line 67: @@
 |&nbsp;&nbsp; &nbsp; &nbsp; The system has only one solution when &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; is any real number except &nbsp;<math style="vertical-align: -13px">\frac{5}{3}.</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: @@
 |To guarantee a unique solution, our matrix must contain two pivots.
 |-
-|So, we must have &nbsp;<math>-3k+5\ne 0.</math>
+|So, we must have &nbsp;<math style="vertical-align: -5px">-3k+5\ne 0.</math>
 |-
 |Hence, we must have
@@ Line 59: / Line 59: @@
 ::<math>k\ne \frac {5}{3}.</math>
 |-
-|Therefore, &nbsp;<math>k</math>&nbsp; can be any real number except &nbsp;<math>\frac{5}{3}.</math>
+|Therefore, &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; can be any real number except &nbsp;<math style="vertical-align: -13px">\frac{5}{3}.</math>
 |}
@@ Line 66: / Line 66: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; &nbsp; &nbsp; The system has only one solution when &nbsp;<math>k</math>&nbsp; is any real number except &nbsp;<math>\frac{5}{3}.</math>
+|&nbsp;&nbsp; &nbsp; &nbsp; The system has only one solution when &nbsp;<math style="vertical-align: 0px">k</math>&nbsp; is any real number except &nbsp;<math style="vertical-align: -13px">\frac{5}{3}.</math>
 |}
 [[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 33: / Line 33: @@
 & k & 1 \\
 & 5  & 2k
           \end{array}\right].</math>
 |}
@@ Line 38: / Line 49: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 2: &nbsp;
 |-
+|
 |}
@@ Line 46: / Line 66: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; &nbsp; &nbsp;
+|&nbsp;&nbsp; &nbsp; &nbsp; The system has only one solution when &nbsp;<math>k</math>&nbsp; is any real number except &nbsp;<math>\frac{5}{3}.</math>
 |}
 [[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 30: / Line 30: @@
 |-
+|
-::<math>B=
+::<math>\left[\begin{array}{cc|c}
 & k & 1 \\
 & 5  & 2k
-          \end{bmatrix}.</math>
+          \end{array}\right].</math>
 |}

@@ Line 1: / Line 1: @@
-<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=
+<span class="exam">Consider the following system of equations.
-::<math>B=
+::<math>x_1+kx_2=1</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>
+::<math>3x_1+5x_2=2k</math>

@@ Line 24: / Line 24: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
 |-
+|
 |}

@@ Line 10: / Line 10: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Foundations: &nbsp;
 |-
+|
 |}