Difference between revisions of "031 Review Part 2, Problem 1"

From Grad Wiki
Jump to navigation Jump to search
(Created page with "<span class="exam">Consider the matrix  <math style="vertical-align: -31px">A= \begin{bmatrix} 1 & -4 & 9 & -7 \\ -1 & 2 & -4 & 1 \\...")
 
 
(6 intermediate revisions by the same user not shown)
Line 16: Line 16:
  
 
<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>
 
<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>
−
 
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 +
|-
 +
|'''1.''' For a matrix &nbsp;<math style="vertical-align: -4px">A,</math>&nbsp; the rank of &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; is
 
|-
 
|-
 
|
 
|
 +
::<math>\text{rank }A=\text{dim Col }A.</math>
 +
|-
 +
|'''2.''' &nbsp;<math style="vertical-align: -1px">\text{Col }A</math>&nbsp; is the vector space spanned by the columns of &nbsp;<math style="vertical-align: 0px">A.</math>
 +
|-
 +
|'''3.''' &nbsp;<math style="vertical-align: -1px">\text{Nul }A</math>&nbsp; is the vector space containing all solutions to &nbsp;<math style="vertical-align: 0px">Ax=0.</math>
 
|}
 
|}
  
Line 31: Line 37:
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|From the matrix &nbsp;<math style="vertical-align: -4px">B,</math>&nbsp; we see that &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; contains two pivots.
 +
|-
 +
|Therefore,
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\text{rank }A} & = & \displaystyle{\text{dim Col }A}\\
 +
&&\\
 +
& = & \displaystyle{2.}
 +
\end{array}</math>
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 +
|-
 +
|By the Rank Theorem, we have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{4} & = & \displaystyle{\text{rank }A+\text{dim Nul }A}\\
 +
&&\\
 +
& = & \displaystyle{2+\text{dim Nul }A.}
 +
\end{array}</math>
 +
|-
 +
|Hence, &nbsp;<math style="vertical-align: -2px">\text{dim Nul }A=2.</math>
 
|}
 
|}
  
Line 45: Line 69:
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|From the matrix &nbsp;<math style="vertical-align: -4px">B,</math>&nbsp; we see that &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; contains pivots in Column 1 and 2.
 +
|-
 +
|So, to obtain a basis for &nbsp;<math style="vertical-align: -4px">\text{Col }A,</math>&nbsp; we select the corresponding columns from &nbsp;<math style="vertical-align: 0px">A.</math>
 +
|-
 +
|Hence, a basis for &nbsp;<math style="vertical-align: -1px">\text{Col }A</math>&nbsp; is
 
|-
 
|-
 
|
 
|
 +
::<math>\Bigg\{\begin{bmatrix}
 +
          1  \\
 +
          -1 \\
 +
          5
 +
        \end{bmatrix},
 +
        \begin{bmatrix}
 +
          -4  \\
 +
          2 \\
 +
          -6
 +
        \end{bmatrix}\Bigg\}.
 +
        </math>
 +
 
|}
 
|}
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 +
|-
 +
|To find a basis for &nbsp;<math style="vertical-align: -1px">\text{Nul }A</math>&nbsp; we translate the matrix equation &nbsp;<math style="vertical-align: -1px">Bx=0</math>&nbsp; back into a system of equations
 +
|-
 +
|and solve for the pivot variables.
 +
|-
 +
|Hence, we have
 +
|-
 +
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{x_1-x_3+5x_4} & = & \displaystyle{0}\\
 +
&&\\
 +
\displaystyle{-2x_2+5x_3-6x_4} & = & \displaystyle{0.}
 +
\end{array}</math>
 +
|-
 +
|Solving for the pivot variables, we have
 +
|-
 +
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{x_1} & = & \displaystyle{x_3-5x_4}\\
 +
&&\\
 +
\displaystyle{x_2} & = & \displaystyle{\frac{5}{2}x_3-3x_4.}
 +
\end{array}</math>
 +
|-
 +
|Hence, the solutions to &nbsp;<math style="vertical-align: -1px">Ax=0</math>&nbsp; are of the form
 +
|-
 +
|
 +
::<math>x_3\begin{bmatrix}
 +
          1  \\
 +
          \frac{5}{2} \\
 +
          1 \\
 +
          0
 +
        \end{bmatrix}+x_4\begin{bmatrix}
 +
          -5  \\
 +
          -3 \\
 +
          0 \\
 +
          1
 +
        \end{bmatrix}.</math>
 +
|-
 +
|Therefore, a basis for &nbsp;<math style="vertical-align: -1px">\text{Nul }A</math>&nbsp; is
 
|-
 
|-
 
|
 
|
 +
::<math>\Bigg\{\begin{bmatrix}
 +
          1  \\
 +
          \frac{5}{2} \\
 +
          1 \\
 +
          0
 +
        \end{bmatrix},
 +
        \begin{bmatrix}
 +
          -5  \\
 +
          -3 \\
 +
          0 \\
 +
          1
 +
        \end{bmatrix}\Bigg\}.
 +
        </math>
 
|}
 
|}
  
Line 59: Line 153:
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math style="vertical-align: -2px">\text{rank }A=2</math>&nbsp; and &nbsp;<math style="vertical-align: -2px">\text{dim Nul }A=2</math>
 +
|-
 +
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; A basis for &nbsp;<math style="vertical-align: -1px">\text{Col }A</math>&nbsp; is &nbsp;<math>\Bigg\{\begin{bmatrix}
 +
          1  \\
 +
          -1 \\
 +
          5
 +
        \end{bmatrix},
 +
        \begin{bmatrix}
 +
          -4  \\
 +
          2 \\
 +
          -6
 +
        \end{bmatrix}\Bigg\}
 +
        </math>
 
|-
 
|-
−
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
+
|&nbsp; &nbsp; &nbsp; &nbsp; and a basis for &nbsp;<math style="vertical-align: -1px">\text{Nul }A</math>&nbsp; is &nbsp;<math>\Bigg\{\begin{bmatrix}
 +
          1  \\
 +
          \frac{5}{2} \\
 +
          1 \\
 +
          0
 +
        \end{bmatrix},
 +
        \begin{bmatrix}
 +
          -5  \\
 +
          -3 \\
 +
          0 \\
 +
          1
 +
        \end{bmatrix}\Bigg\}.
 +
        </math>
 
|}
 
|}
−
[[009C_Sample_Final_1|'''<u>Return to Sample Exam</u>''']]
+
[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

Latest revision as of 13:12, 15 October 2017

Consider the matrix    and assume that it is row equivalent to the matrix

(a) List rank    and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{dim Nul }A.}

(b) Find bases for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Col }A}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Nul }A.}   Find an example of a nonzero vector that belongs to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Col }A,}   as well as an example of a nonzero vector that belongs to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Nul }A.}

Foundations:  
1. For a matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A,}   the rank of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{rank }A=\text{dim Col }A.}
2.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Col }A}   is the vector space spanned by the columns of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A.}
3.  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Nul }A}   is the vector space containing all solutions to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ax=0.}


Solution:

(a)

Step 1:  
From the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B,}   we see that  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   contains two pivots.
Therefore,

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\text{rank }A} & = & \displaystyle{\text{dim Col }A}\\ &&\\ & = & \displaystyle{2.} \end{array}}

Step 2:  
By the Rank Theorem, we have

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{4} & = & \displaystyle{\text{rank }A+\text{dim Nul }A}\\ &&\\ & = & \displaystyle{2+\text{dim Nul }A.} \end{array}}

Hence,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{dim Nul }A=2.}

(b)

Step 1:  
From the matrix  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B,}   we see that  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}   contains pivots in Column 1 and 2.
So, to obtain a basis for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Col }A,}   we select the corresponding columns from  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A.}
Hence, a basis for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Col }A}   is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Bigg\{\begin{bmatrix} 1 \\ -1 \\ 5 \end{bmatrix}, \begin{bmatrix} -4 \\ 2 \\ -6 \end{bmatrix}\Bigg\}. }
Step 2:  
To find a basis for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Nul }A}   we translate the matrix equation  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Bx=0}   back into a system of equations
and solve for the pivot variables.
Hence, we have

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{x_1-x_3+5x_4} & = & \displaystyle{0}\\ &&\\ \displaystyle{-2x_2+5x_3-6x_4} & = & \displaystyle{0.} \end{array}}

Solving for the pivot variables, we have

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{x_1} & = & \displaystyle{x_3-5x_4}\\ &&\\ \displaystyle{x_2} & = & \displaystyle{\frac{5}{2}x_3-3x_4.} \end{array}}

Hence, the solutions to  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ax=0}   are of the form
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_3\begin{bmatrix} 1 \\ \frac{5}{2} \\ 1 \\ 0 \end{bmatrix}+x_4\begin{bmatrix} -5 \\ -3 \\ 0 \\ 1 \end{bmatrix}.}
Therefore, a basis for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Nul }A}   is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Bigg\{\begin{bmatrix} 1 \\ \frac{5}{2} \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -5 \\ -3 \\ 0 \\ 1 \end{bmatrix}\Bigg\}. }


Final Answer:  
   (a)     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{rank }A=2}   and  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{dim Nul }A=2}
   (b)     A basis for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Col }A}   is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Bigg\{\begin{bmatrix} 1 \\ -1 \\ 5 \end{bmatrix}, \begin{bmatrix} -4 \\ 2 \\ -6 \end{bmatrix}\Bigg\} }
        and a basis for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Nul }A}   is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Bigg\{\begin{bmatrix} 1 \\ \frac{5}{2} \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -5 \\ -3 \\ 0 \\ 1 \end{bmatrix}\Bigg\}. }

Return to Review Problems

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=031_Review_Part_2,_Problem_1&oldid=3659"