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

From Grad Wiki
Jump to navigation Jump to search
(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 \\...")
 
Line 1: Line 1:
−
<span class="exam">Consider the matrix &nbsp;<math style="vertical-align: -31px">A=   
+
<span class="exam"> Suppose &nbsp;<math style="vertical-align: 0px">T</math>&nbsp; is a linear transformation given by the formula
−
    \begin{bmatrix}
 
−
          1 & -4 & 9 & -7 \\
 
−
          -1 & 2  & -4 & 1 \\
 
−
          5 & -6 & 10 & 7
 
−
        \end{bmatrix}</math>&nbsp; and assume that it is row equivalent to the matrix
 
  
−
::<math>B=   
+
::<math>T\Bigg(
−
    \begin{bmatrix}
+
\begin{bmatrix}
−
           1 & 0 & -1 & 5 \\
+
           x_1 \\
−
           0 & -2 & 5 & -6 \\
+
          x_2 \\
−
           0 & 0 & 0 & 0
+
          x_3 \\
−
         \end{bmatrix}.</math>    
+
        \end{bmatrix}
−
   
+
        \Bigg)=
−
<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>
+
\begin{bmatrix}
−
 
+
           5x_1-2.5x_2+10x_3 \\
−
<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>
+
           -x_1+0.5x_2-2x_3
 +
         \end{bmatrix}</math>
 +
       
 +
<span class="exam">(a) Find the standard matrix for &nbsp;<math style="vertical-align: 0px">T.</math>
 +
       
 +
<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>
 +
       
 +
<span class="exam">(c) Is &nbsp;<math style="vertical-align: -21px">\begin{bmatrix}
 +
          -1 \\
 +
          3
 +
        \end{bmatrix}</math>&nbsp; in the range of &nbsp;<math style="vertical-align: 0px">T?</math>&nbsp; Explain.
  
  
Line 42: Line 47:
  
 
'''(b)'''
 
'''(b)'''
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Step 1: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 +
!Step 2: &nbsp;
 +
|-
 +
|
 +
|}
 +
 +
'''(c)'''
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"

Revision as of 18:09, 9 October 2017

Suppose    is a linear transformation given by the formula

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 T\Bigg( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \end{bmatrix} \Bigg)= \begin{bmatrix} 5x_1-2.5x_2+10x_3 \\ -x_1+0.5x_2-2x_3 \end{bmatrix}}

(a) Find the standard matrix for  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 T.}

(b) Let  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 \vec{u}=7\vec{e_1}-4\vec{e_2}.}   Find  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 T(\vec{u}).}

(c) Is  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 \begin{bmatrix} -1 \\ 3 \end{bmatrix}}   in the range 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 T?}   Explain.


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

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=031_Review_Part_2,_Problem_4&oldid=3560"