Difference between revisions of "031 Review Part 2, Problem 6"

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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"> Let &nbsp;<math>\vec{v}=\begin{bmatrix}
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    \begin{bmatrix}
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           -1 \\
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           1 & -4 & 9 & -7 \\
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           3 \\
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           -1 & 2  & -4 & 1 \\
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           0
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           5 & -6 & 10 & 7
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         \end{bmatrix}</math>&nbsp; and &nbsp;<math>\vec{y}=\begin{bmatrix}
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         \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
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           2 \\
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           0 \\
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::<math>B=  
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           5
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    \begin{bmatrix}
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         \end{bmatrix}.</math>
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           1 & 0 & -1 & 5 \\
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           0 & -2  & 5 & -6 \\
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<span class="exam">(a) Find a unit vector in the direction of &nbsp;<math style="vertical-align: 0px">\vec{v}.</math>
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           0 & 0 & 0 & 0
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         \end{bmatrix}.</math>    
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<span class="exam">(b) Find the distance between &nbsp;<math style="vertical-align: 0px">\vec{v}</math>&nbsp; and &nbsp;<math style="vertical-align: -3px">\vec{y}.</math>
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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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<span class="exam">(c) Let &nbsp;<math style="vertical-align: -5px">L=\text{Span }\{\vec{v}\}.</math>&nbsp; Compute the orthogonal projection of &nbsp;<math style="vertical-align: -3px">\vec{y}</math>&nbsp; onto &nbsp;<math style="vertical-align: 0px">L.</math>
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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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'''(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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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 2: &nbsp;
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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 19:10, 9 October 2017

Let    and  

(a) Find a unit vector in the direction of  

(b) Find the distance between    and  

(c) Let    Compute the orthogonal projection of    onto  


Foundations:  


Solution:

(a)

Step 1:  
Step 2:  

(b)

Step 1:  
Step 2:  

(c)

Step 1:  
Step 2:  


Final Answer:  
   (a)    
   (b)    

Return to Sample Exam

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