Difference between revisions of "031 Review Part 3, Problem 11"

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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: -5px">\{\vec{u},\vec{v}\}</math>&nbsp; is a basis of the eigenspace corresponding to the eigenvalue 0 of a &nbsp;<math style="vertical-align: 0px">5\times 5</math>&nbsp; matrix &nbsp;<math style="vertical-align: 0px">A.</math>
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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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<span class="exam">(a) Is &nbsp;<math style="vertical-align: 0px">\vec{w}=\vec{u}-2\vec{v}</math>&nbsp; an eigenvector of &nbsp;<math style="vertical-align: 0px">A?</math>&nbsp; If so, find the corresponding eigenvalue.
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    \begin{bmatrix}
 
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          1 & 0 & -1 & 5 \\
 
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          0 & -2  & 5 & -6 \\
 
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          0 & 0 & 0 & 0
 
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        \end{bmatrix}.</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">(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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<span class="exam">If not, explain why.
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<span class="exam">(b) Find the dimension of &nbsp;<math style="vertical-align: -1px">\text{Col }A.</math>
  
  

Revision as of 19:32, 9 October 2017

Suppose    is a basis of the eigenspace corresponding to the eigenvalue 0 of a    matrix  

(a) Is    an eigenvector of    If so, find the corresponding eigenvalue.

If not, explain why.

(b) Find the dimension of  


Foundations:  


Solution:

(a)

Step 1:  
Step 2:  

(b)

Step 1:  
Step 2:  


Final Answer:  
   (a)    
   (b)    

Return to Sample Exam

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