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

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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 \\...")
 
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<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=   
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<span class="exam"> Suppose &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation given by the formula
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    \begin{bmatrix}
 
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          1 & -4 & 9 & -7 \\
 
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          -1 & 2  & -4 & 1 \\
 
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          5 & -6 & 10 & 7
 
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        \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
 
  
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::<math>B=   
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::<math>T\Bigg(
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    \begin{bmatrix}
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\begin{bmatrix}
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           1 & 0 & -1 & 5 \\
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           x_1 \\
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           0 & -2 & 5 & -6 \\
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          x_2 \\
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           0 & 0 & 0 & 0
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          x_3 \\
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         \end{bmatrix}.</math>    
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        \end{bmatrix}
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        \Bigg)=
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<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>
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\begin{bmatrix}
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           5x_1-2.5x_2+10x_3 \\
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<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>
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           -x_1+0.5x_2-2x_3
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         \end{bmatrix}</math>
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<span class="exam">(a) Find the standard matrix for &nbsp;<math style="vertical-align: 0px">T.</math>
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<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>
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<span class="exam">(c) Is &nbsp;<math style="vertical-align: -21px">\begin{bmatrix}
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          -1 \\
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          3
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        \end{bmatrix}</math>&nbsp; in the range of &nbsp;<math style="vertical-align: 0px">T?</math>&nbsp; Explain.
  
  
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'''(b)'''
 
'''(b)'''
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 1: &nbsp;
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|-
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|
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|}
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 2: &nbsp;
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|-
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|
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|}
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'''(c)'''
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

Revision as of 19:09, 9 October 2017

Suppose    is a linear transformation given by the formula

(a) Find the standard matrix for  

(b) Let    Find  

(c) Is    in the range of    Explain.


Foundations:  


Solution:

(a)

Step 1:  
Step 2:  

(b)

Step 1:  
Step 2:  

(c)

Step 1:  
Step 2:  


Final Answer:  
   (a)    
   (b)    

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