Difference between revisions of "031 Review Part 2, Problem 7"

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(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 \\...")
 
 
(4 intermediate revisions by the same user not shown)
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<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=   
+
<span class="exam">(a) Let &nbsp;<math style="vertical-align: -2px">T:\mathbb{R}^2\rightarrow \mathbb{R}^2</math>&nbsp; be a transformation given by
−
    \begin{bmatrix}
 
−
          1 & -4 & 9 & -7 \\
 
−
          -1 & 2 & -4 & 1 \\
 
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          5 & -6 & 10 & 7
 
−
        \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
 
  
−
::<math>B=   
+
::<math>T\bigg(
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    \begin{bmatrix}
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\begin{bmatrix}
−
           1 & 0 & -1 & 5 \\
+
           x \\
−
           0 & -2  & 5 & -6 \\
+
           y
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          0 & 0 & 0 & 0
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         \end{bmatrix}
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         \end{bmatrix}.</math>     
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        \bigg)=
−
   
+
\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>
+
          1-xy \\
 +
          x+y
 +
        \end{bmatrix}.</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>
+
<span class="exam">Determine whether &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation. Explain.
  
 +
<span class="exam">(b) Let &nbsp;<math style="vertical-align: -19px">A=   
 +
    \begin{bmatrix}
 +
          1 & -3 & 0 \\
 +
          -4 & 1 &1
 +
        \end{bmatrix}</math>&nbsp; and &nbsp;<math style="vertical-align: -32px">B=   
 +
    \begin{bmatrix}
 +
          2 & 1\\
 +
          1 & 0 \\
 +
          -1 & 1
 +
        \end{bmatrix}.</math>&nbsp; Find &nbsp;<math style="vertical-align: -4px">AB,~BA^T</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">A-B^T.</math>
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 +
|-
 +
|A map &nbsp;<math style="vertical-align: -2px">T:\mathbb{R}^n\rightarrow \mathbb{R}^m</math>&nbsp; is a linear transformation if
 
|-
 
|-
 
|
 
|
 +
::<math>T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})</math>
 +
|-
 +
|
 +
:and
 +
|-
 +
|
 +
::<math>T(a\vec{x})=aT(\vec{x})</math>
 +
|-
 +
|
 +
:for all &nbsp;<math style="vertical-align: -4px">\vec{x},\vec{y}\in \mathbb{R}^n</math>&nbsp; and all &nbsp;<math style="vertical-align: -1px">a\in \mathbb{R}.</math>
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|We claim that &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is not a linear transformation.
 +
|-
 +
|Consider the vectors &nbsp;<math style="vertical-align: -20px">\begin{bmatrix}
 +
          1\\
 +
          0
 +
        \end{bmatrix}</math>&nbsp; and &nbsp;<math style="vertical-align: -20px">\begin{bmatrix}
 +
          0\\
 +
          1
 +
        \end{bmatrix}.</math>
 +
|-
 +
|Then, we have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{T\bigg(\begin{bmatrix}
 +
          1\\
 +
          0
 +
        \end{bmatrix}+\begin{bmatrix}
 +
          0\\
 +
          1
 +
        \end{bmatrix}\bigg)} & = & \displaystyle{T\bigg(\begin{bmatrix}
 +
          1\\
 +
          1
 +
        \end{bmatrix}\bigg)}\\
 +
&&\\
 +
& = & \displaystyle{\begin{bmatrix}
 +
          0\\
 +
          2
 +
        \end{bmatrix}.}
 +
\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;
 +
|-
 +
|On the other hand, notice
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{T\bigg(\begin{bmatrix}
 +
          1\\
 +
          0
 +
        \end{bmatrix}\bigg)+T\bigg(\begin{bmatrix}
 +
          0\\
 +
          1
 +
        \end{bmatrix}\bigg)} & = & \displaystyle{\begin{bmatrix}
 +
          1\\
 +
          1
 +
        \end{bmatrix}+\begin{bmatrix}
 +
          1\\
 +
          1
 +
        \end{bmatrix}}\\
 +
&&\\
 +
& = & \displaystyle{\begin{bmatrix}
 +
          2\\
 +
          2
 +
        \end{bmatrix}.}
 +
\end{array}</math>
 +
|-
 +
|So, now we know
 +
|-
 +
|
 +
::<math>T\bigg(\begin{bmatrix}
 +
          1\\
 +
          0
 +
        \end{bmatrix}+\begin{bmatrix}
 +
          0\\
 +
          1
 +
        \end{bmatrix}\bigg)\neq T\bigg(\begin{bmatrix}
 +
          1\\
 +
          0
 +
        \end{bmatrix}\bigg)+T\bigg(\begin{bmatrix}
 +
          0\\
 +
          1
 +
        \end{bmatrix}\bigg).</math>
 +
|-
 +
|Therefore, &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is not a linear transformation.
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|Using the row-column rule for multiplication, we have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{AB} & = & \displaystyle{\begin{bmatrix}
 +
          1 & -3 & 0 \\
 +
          -4 & 1 &1
 +
        \end{bmatrix}\begin{bmatrix}
 +
          2 & 1\\
 +
          1 & 0 \\
 +
          -1 & 1
 +
        \end{bmatrix}}\\
 +
&&\\
 +
& = & \displaystyle{\begin{bmatrix}
 +
          1(2)+-3(1)+0(-1) & 1(1)+-3(0)+0(1)\\
 +
          -4(2)+1(1)+1(-1) & -4(1)+1(0)+1(1)
 +
        \end{bmatrix}}\\
 +
&&\\
 +
& = & \displaystyle{\begin{bmatrix}
 +
          -1 & 1\\
 +
          -8 & -3
 +
        \end{bmatrix}.}
 +
\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;
 +
|-
 +
|Now, &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">A^T</math>&nbsp; are both &nbsp;<math style="vertical-align: 0px">3\times 2</math>&nbsp; matrices.
 +
|-
 +
|Hence, &nbsp;<math style="vertical-align: 0px">BA^T</math>&nbsp; is undefined.
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Step 3: &nbsp;
 +
|-
 +
|For &nbsp;<math style="vertical-align: -5px">A-B^T,</math>&nbsp; we have
 
|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{A-B^T} & = & \displaystyle{\begin{bmatrix}
 +
          1 & -3 & 0 \\
 +
          -4 & 1 &1
 +
        \end{bmatrix}-\begin{bmatrix}
 +
          2 & 1 & -1\\
 +
          1 & 0 & 1\\
 +
        \end{bmatrix}}\\
 +
&&\\
 +
& = & \displaystyle{\begin{bmatrix}
 +
          -1 & -4 & 1\\
 +
          -5 & 1 & 0
 +
        \end{bmatrix}.}
 +
\end{array}</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math style="vertical-align: 0px">T</math>&nbsp; is not a linear transformation
 
|-
 
|-
−
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math style="vertical-align: -20px">AB=\begin{bmatrix}
 +
          -1 & 1\\
 +
          -8 & -3
 +
        \end{bmatrix},</math>&nbsp; <math style="vertical-align: -1px">BA^T</math>&nbsp; is undefined and &nbsp;<math style="vertical-align: -20px">A-B^T=\begin{bmatrix}
 +
          -1 & -4 & 1\\
 +
          -5 & 1 & 0
 +
        \end{bmatrix}</math>
 
|}
 
|}
−
[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
+
[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

Latest revision as of 12:36, 15 October 2017

(a) Let  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 T:\mathbb{R}^2\rightarrow \mathbb{R}^2}   be a transformation given by

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 T\bigg( \begin{bmatrix} x \\ y \end{bmatrix} \bigg)= \begin{bmatrix} 1-xy \\ x+y \end{bmatrix}.}

Determine whether  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 T}   is a linear transformation. Explain.

(b) Let  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 & -3 & 0 \\ -4 & 1 &1 \end{bmatrix}}   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 B= \begin{bmatrix} 2 & 1\\ 1 & 0 \\ -1 & 1 \end{bmatrix}.}   Find  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 AB,~BA^T}   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 A-B^T.}

Foundations:  
A map  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 T:\mathbb{R}^n\rightarrow \mathbb{R}^m}   is a linear transformation if
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 T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})}
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 T(a\vec{x})=aT(\vec{x})}
for all  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 \vec{x},\vec{y}\in \mathbb{R}^n}   and all  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\in \mathbb{R}.}


Solution:

(a)

Step 1:  
We claim 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 T}   is not a linear transformation.
Consider the vectors  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{bmatrix} 1\\ 0 \end{bmatrix}}   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 \begin{bmatrix} 0\\ 1 \end{bmatrix}.}
Then, 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{T\bigg(\begin{bmatrix} 1\\ 0 \end{bmatrix}+\begin{bmatrix} 0\\ 1 \end{bmatrix}\bigg)} & = & \displaystyle{T\bigg(\begin{bmatrix} 1\\ 1 \end{bmatrix}\bigg)}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 0\\ 2 \end{bmatrix}.} \end{array}}

Step 2:  
On the other hand, notice

       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{T\bigg(\begin{bmatrix} 1\\ 0 \end{bmatrix}\bigg)+T\bigg(\begin{bmatrix} 0\\ 1 \end{bmatrix}\bigg)} & = & \displaystyle{\begin{bmatrix} 1\\ 1 \end{bmatrix}+\begin{bmatrix} 1\\ 1 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 2\\ 2 \end{bmatrix}.} \end{array}}

So, now we know
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 T\bigg(\begin{bmatrix} 1\\ 0 \end{bmatrix}+\begin{bmatrix} 0\\ 1 \end{bmatrix}\bigg)\neq T\bigg(\begin{bmatrix} 1\\ 0 \end{bmatrix}\bigg)+T\bigg(\begin{bmatrix} 0\\ 1 \end{bmatrix}\bigg).}
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 T}   is not a linear transformation.

(b)

Step 1:  
Using the row-column rule for multiplication, 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{AB} & = & \displaystyle{\begin{bmatrix} 1 & -3 & 0 \\ -4 & 1 &1 \end{bmatrix}\begin{bmatrix} 2 & 1\\ 1 & 0 \\ -1 & 1 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 1(2)+-3(1)+0(-1) & 1(1)+-3(0)+0(1)\\ -4(2)+1(1)+1(-1) & -4(1)+1(0)+1(1) \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} -1 & 1\\ -8 & -3 \end{bmatrix}.} \end{array}}

Step 2:  
Now,  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}   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 A^T}   are both  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 3\times 2}   matrices.
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 BA^T}   is undefined.
Step 3:  
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 A-B^T,}   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{A-B^T} & = & \displaystyle{\begin{bmatrix} 1 & -3 & 0 \\ -4 & 1 &1 \end{bmatrix}-\begin{bmatrix} 2 & 1 & -1\\ 1 & 0 & 1\\ \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} -1 & -4 & 1\\ -5 & 1 & 0 \end{bmatrix}.} \end{array}}


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 T}   is not a linear transformation
   (b)     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 AB=\begin{bmatrix} -1 & 1\\ -8 & -3 \end{bmatrix},}   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 BA^T}   is undefined 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 A-B^T=\begin{bmatrix} -1 & -4 & 1\\ -5 & 1 & 0 \end{bmatrix}}

Return to Review Problems

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