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

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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 2:  
 
!Step 2:  
 +
|-
 +
|To find a basis for &nbsp;<math>\text{Nul }A</math> we translate the matrix equation &nbsp;<math>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>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>
 +
|-
 +
|Hence, 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>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math>\text{rank }A=2</math>&nbsp; and <math style="vertical-align: -2px">\text{dim Nul }A=2.</math>
 +
|-
 +
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; A basis for &nbsp;<math>\text{Col }A</math>&nbsp; is <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>\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>
 
|}
 
|}
 
[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
 
[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]

Revision as of 17:23, 10 October 2017

Consider 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 A= \begin{bmatrix} 1 & -4 & 9 & -7 \\ -1 & 2 & -4 & 1 \\ 5 & -6 & 10 & 7 \end{bmatrix}}   and assume that it is row equivalent to 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= \begin{bmatrix} 1 & 0 & -1 & 5 \\ 0 & -2 & 5 & -6 \\ 0 & 0 & 0 & 0 \end{bmatrix}.}

(a) List rank  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}   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}.}
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{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 Sample Exam

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