Difference between revisions of "031 Review Part 2, 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 
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&nbsp;<math>B=     
 
     \begin{bmatrix}
 
     \begin{bmatrix}
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           1 & -4 & 9 & -7 \\
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           1 & -2 & 3 & 4\\
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           -1 & 2 & -4 & 1 \\
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          0 & 3 &0 &0\\
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           5 & -6 & 10 & 7
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           0 & 5 & 1 & 2\\
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         \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
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           0 & -1 & 3 & 6
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         \end{bmatrix}.
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</math>
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<span class="exam">(a) Is &nbsp;<math style="vertical-align: 0px">B</math>&nbsp; invertible? Explain.
  
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::<math>B=   
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<span class="exam">(b) Define a linear transformation &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; by the formula &nbsp;<math style="vertical-align: -5px">T(\vec{x})=B\vec{x}.</math>&nbsp; Is &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; onto? Explain.
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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>
 
  
  

Revision as of 19:08, 9 October 2017

Let  

(a) Is    invertible? Explain.

(b) Define a linear transformation    by the formula    Is    onto? Explain.


Foundations:  


Solution:

(a)

Step 1:  
Step 2:  

(b)

Step 1:  
Step 2:  


Final Answer:  
   (a)    
   (b)    

Return to Sample Exam

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