Difference between revisions of "031 Review Part 2, Problem 4"

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 \\...")
 
 
(5 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=   
+
<span class="exam"> Suppose &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation given by the formula
−
    \begin{bmatrix}
 
−
          1 & -4 & 9 & -7 \\
 
−
          -1 & 2  & -4 & 1 \\
 
−
          5 & -6 & 10 & 7
 
−
        \end{bmatrix}</math>&nbsp; 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>     
 
−
   
 
−
<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>
 
  
 +
::<math>T\Bigg(
 +
\begin{bmatrix}
 +
          x_1 \\
 +
          x_2 \\
 +
          x_3 \\
 +
        \end{bmatrix}
 +
        \Bigg)=
 +
\begin{bmatrix}
 +
          5x_1-2.5x_2+10x_3 \\
 +
          -x_1+0.5x_2-2x_3
 +
        \end{bmatrix}</math>
 +
       
 +
<span class="exam">(a) Find the standard matrix for &nbsp;<math style="vertical-align: 0px">T.</math>
 +
       
 +
<span class="exam">(b) Let &nbsp;<math style="vertical-align: -5px">\vec{u}=7\vec{e_1}-4\vec{e_2}.</math>&nbsp; Find &nbsp;<math style="vertical-align: -6px">T(\vec{u}).</math>
 +
       
 +
<span class="exam">(c) Is &nbsp;<math style="vertical-align: -21px">\begin{bmatrix}
 +
          -1 \\
 +
          3
 +
        \end{bmatrix}</math>&nbsp; in the range of &nbsp;<math style="vertical-align: 0px">T?</math>&nbsp; Explain.
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
 +
|-
 +
|'''1.''' The standard matrix of a linear transformation &nbsp;<math style="vertical-align: -2px">T:\mathbb{R}^n\rightarrow \mathbb{R}^m</math>&nbsp; is given by
 +
|-
 +
|
 +
::<math>\begin{bmatrix}
 +
          T(\vec{e_1}) & T(\vec{e_2}) & \cdots & T(\vec{e_n})
 +
        \end{bmatrix}
 +
</math>
 +
|-
 +
|
 +
:where &nbsp;<math style="vertical-align: -5px">\{e_1,e_2,\ldots,e_n\}</math>&nbsp; is the standard basis of &nbsp;<math style="vertical-align: -1px">\mathbb{R}^n.</math>
 +
|-
 +
|'''2.''' A vector &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; is in the image of &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; if there exists &nbsp;<math style="vertical-align: 0px">\vec{x}</math>&nbsp; such that
 
|-
 
|-
 
|
 
|
 +
::<math>T(\vec{x})=\vec{v}.</math>
 
|}
 
|}
  
Line 31: Line 49:
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|Notice, we have
 
|-
 
|-
 
|
 
|
 +
::<math>T(\vec{e_1})=
 +
\begin{bmatrix}
 +
          5 \\
 +
          -1
 +
        \end{bmatrix},~T(\vec{e_2})=
 +
\begin{bmatrix}
 +
          -2.5 \\
 +
          0.5
 +
        \end{bmatrix},\text{ and }T(\vec{e_3})=
 +
\begin{bmatrix}
 +
          10 \\
 +
          -2
 +
        \end{bmatrix}.</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;
 +
|-
 +
|So, the standard matrix of &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is
 
|-
 
|-
 
|
 
|
 +
::<math>[T]=\begin{bmatrix}
 +
          5 & -2.5 &10 \\
 +
          -1 & 0.5 & -2
 +
        \end{bmatrix}.</math>
 
|}
 
|}
  
Line 45: Line 84:
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 +
|-
 +
|Since &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation, we know
 +
|-
 +
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{T(\vec{u})} & = & \displaystyle{T(7\vec{e_1}-4\vec{e_2})}\\
 +
&&\\
 +
& = & \displaystyle{T(7\vec{e_1})-T(4\vec{e_2})}\\
 +
&&\\
 +
& = & \displaystyle{7T(\vec{e_1})-4T(\vec{e_2}).}
 +
\end{array}</math>
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Step 2: &nbsp;
 +
|-
 +
|Now, we have
 +
|-
 +
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{T(\vec{u})} & = & \displaystyle{7\begin{bmatrix}
 +
          5 \\
 +
          -1
 +
        \end{bmatrix}-4\begin{bmatrix}
 +
          -2.5 \\
 +
          0.5
 +
        \end{bmatrix}}\\
 +
&&\\
 +
& = & \displaystyle{\begin{bmatrix}
 +
          35 \\
 +
          -7
 +
        \end{bmatrix}+\begin{bmatrix}
 +
          10 \\
 +
          -2
 +
        \end{bmatrix}}\\
 +
&&\\
 +
& = & \displaystyle{\begin{bmatrix}
 +
          45 \\
 +
          -9
 +
        \end{bmatrix}.}
 +
\end{array}</math>
 +
|}
 +
 +
'''(c)'''
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Step 1: &nbsp;
 +
|-
 +
|To answer this question, we augment the standard matrix of &nbsp;<math style="vertical-align: -1px">T</math>&nbsp; with this vector and row reduce this matrix.
 +
|-
 +
|So, we have the matrix
 
|-
 
|-
 
|
 
|
 +
::<math>\left[\begin{array}{ccc|c} 
 +
          5 & -2.5 & 10 & -1\\
 +
          -1 & 0.5  & -2  & 3
 +
        \end{array}\right].</math>
 
|}
 
|}
  
Line 53: Line 147:
 
|-
 
|-
 
|
 
|
 +
Now, row reducing this matrix, we have
 +
|-
 +
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\left[\begin{array}{ccc|c} 
 +
            5 & -2.5 & 10 & -1\\
 +
          -1 & 0.5  & -2  & 3
 +
        \end{array}\right]} & \sim & \displaystyle{\left[\begin{array}{ccc|c} 
 +
          5 & -2.5 & 10 & -1\\
 +
          -5 & 2.5  & -10  & 15
 +
        \end{array}\right]}\\
 +
&&\\
 +
& \sim & \displaystyle{\left[\begin{array}{ccc|c} 
 +
          5 & -2.5 & 10 & -1\\
 +
          0 & 0  & 0  & 14
 +
        \end{array}\right].}
 +
\end{array}</math>
 +
|-
 +
|From here, we can tell that the corresponding system is inconsistent.
 +
|-
 +
|Hence, this vector is not in the range of &nbsp;<math style="vertical-align: 0px">T.</math>&nbsp;
 
|}
 
|}
  
Line 59: Line 174:
 
!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(a)''' &nbsp; &nbsp; <math>[T]=\begin{bmatrix}
 +
          5 & -2.5 &10 \\
 +
          -1 & 0.5 & -2
 +
        \end{bmatrix}</math>
 +
|-
 +
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp; <math>\begin{bmatrix}
 +
          45 \\
 +
          -9
 +
        \end{bmatrix}</math>
 
|-
 
|-
−
|&nbsp;&nbsp; '''(b)''' &nbsp; &nbsp;  
+
|&nbsp;&nbsp; '''(c)''' &nbsp; &nbsp; No, see above
 
|}
 
|}
−
[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
+
[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

Latest revision as of 12:27, 15 October 2017

Suppose  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 given by the formula

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_1 \\ x_2 \\ x_3 \\ \end{bmatrix} \Bigg)= \begin{bmatrix} 5x_1-2.5x_2+10x_3 \\ -x_1+0.5x_2-2x_3 \end{bmatrix}}

(a) Find the standard matrix 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 T.}

(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 \vec{u}=7\vec{e_1}-4\vec{e_2}.}   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 T(\vec{u}).}

(c) 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 \begin{bmatrix} -1 \\ 3 \end{bmatrix}}   in the range 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 T?}   Explain.

Foundations:  
1. The standard matrix of a linear transformation  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 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 \begin{bmatrix} T(\vec{e_1}) & T(\vec{e_2}) & \cdots & T(\vec{e_n}) \end{bmatrix} }
where  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 \{e_1,e_2,\ldots,e_n\}}   is the standard basis 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 \mathbb{R}^n.}
2. A vector  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{v}}   is in the image 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 T}   if there exists  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}}   such 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(\vec{x})=\vec{v}.}


Solution:

(a)

Step 1:  
Notice, 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 T(\vec{e_1})= \begin{bmatrix} 5 \\ -1 \end{bmatrix},~T(\vec{e_2})= \begin{bmatrix} -2.5 \\ 0.5 \end{bmatrix},\text{ and }T(\vec{e_3})= \begin{bmatrix} 10 \\ -2 \end{bmatrix}.}
Step 2:  
So, the standard matrix 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 T}   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 [T]=\begin{bmatrix} 5 & -2.5 &10 \\ -1 & 0.5 & -2 \end{bmatrix}.}

(b)

Step 1:  
Since  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, 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 \begin{array}{rcl} \displaystyle{T(\vec{u})} & = & \displaystyle{T(7\vec{e_1}-4\vec{e_2})}\\ &&\\ & = & \displaystyle{T(7\vec{e_1})-T(4\vec{e_2})}\\ &&\\ & = & \displaystyle{7T(\vec{e_1})-4T(\vec{e_2}).} \end{array}}

Step 2:  
Now, 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(\vec{u})} & = & \displaystyle{7\begin{bmatrix} 5 \\ -1 \end{bmatrix}-4\begin{bmatrix} -2.5 \\ 0.5 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 35 \\ -7 \end{bmatrix}+\begin{bmatrix} 10 \\ -2 \end{bmatrix}}\\ &&\\ & = & \displaystyle{\begin{bmatrix} 45 \\ -9 \end{bmatrix}.} \end{array}}

(c)

Step 1:  
To answer this question, we augment the standard matrix 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 T}   with this vector and row reduce this matrix.
So, we have 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 \left[\begin{array}{ccc|c} 5 & -2.5 & 10 & -1\\ -1 & 0.5 & -2 & 3 \end{array}\right].}
Step 2:  

Now, row reducing this matrix, 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{\left[\begin{array}{ccc|c} 5 & -2.5 & 10 & -1\\ -1 & 0.5 & -2 & 3 \end{array}\right]} & \sim & \displaystyle{\left[\begin{array}{ccc|c} 5 & -2.5 & 10 & -1\\ -5 & 2.5 & -10 & 15 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{ccc|c} 5 & -2.5 & 10 & -1\\ 0 & 0 & 0 & 14 \end{array}\right].} \end{array}}

From here, we can tell that the corresponding system is inconsistent.
Hence, this vector is not in the range 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 T.}  


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]=\begin{bmatrix} 5 & -2.5 &10 \\ -1 & 0.5 & -2 \end{bmatrix}}
   (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 \begin{bmatrix} 45 \\ -9 \end{bmatrix}}
   (c)     No, see above

Return to Review Problems

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=031_Review_Part_2,_Problem_4&oldid=3662"