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

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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 style="vertical-align: -20px">A=     
 
     \begin{bmatrix}
 
     \begin{bmatrix}
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           1 & -4 & 9 & -7 \\
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           5 & 1 \\
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           -1 & 2  & -4 & 1 \\
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           0 & 5  
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          5 & -6 & 10 & 7
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         \end{bmatrix}.</math>
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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) Find a basis for the eigenspace(s) of &nbsp;<math style="vertical-align: 0px">A.</math>
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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">(b) Is the matrix &nbsp;<math style="vertical-align: 0px">A</math>&nbsp; diagonalizable? Explain.
  
  

Revision as of 18:23, 9 October 2017

Let  

(a) Find a basis for the eigenspace(s) 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 A.}

(b) Is 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 A}   diagonalizable? Explain.


Foundations:  


Solution:

(a)

Step 1:  
Step 2:  

(b)

Step 1:  
Step 2:  


Final Answer:  
   (a)    
   (b)    

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