Difference between revisions of "031 Review Part 3, Problem 4"

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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
−
!Step 1:    
+
!Step 1:  
 +
|-
 +
|To determine whether the vector
 
|-
 
|-
 
|
 
|
 +
::<math>\begin{bmatrix}
 +
          2 \\
 +
          6 \\
 +
          4 \\
 +
          0
 +
        \end{bmatrix}</math>
 +
|-
 +
|is in  &nbsp;<math>W^\perp,</math>&nbsp; it suffices to see if this vector is orthogonal to
 +
|-
 +
|the basis elements of &nbsp;<math>W.</math>
 +
|-
 +
|Notice that we have
 +
|-
 +
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\begin{bmatrix}
 +
          2 \\
 +
          6 \\
 +
          4 \\
 +
          0
 +
        \end{bmatrix}\cdot \begin{bmatrix}
 +
          2 \\
 +
          0 \\
 +
          -1 \\
 +
          0
 +
        \end{bmatrix}} & = & \displaystyle{2(2)+6(0)+4(-1)+0(0)}\\
 +
&&\\
 +
& = & \displaystyle{0.}
 +
\end{array}</math>
 
|}
 
|}
  
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|-
 
|-
 
|
 
|
 +
&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
 +
\displaystyle{\begin{bmatrix}
 +
          2 \\
 +
          6 \\
 +
          4 \\
 +
          0
 +
        \end{bmatrix}\cdot \begin{bmatrix}
 +
          -3 \\
 +
          1 \\
 +
          0 \\
 +
          0
 +
        \end{bmatrix}} & = & \displaystyle{2(-3)+6(1)+4(0)+0(0)}\\
 +
&&\\
 +
& = & \displaystyle{0.}
 +
\end{array}</math>
 +
|-
 +
|Hence, we conclude
 +
|-
 +
|
 +
::<math>\begin{bmatrix}
 +
          2 \\
 +
          6 \\
 +
          4 \\
 +
          0
 +
        \end{bmatrix}\in W^\perp.</math>
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
|-
 
|-
−
|&nbsp;&nbsp; &nbsp; &nbsp;  
+
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{bmatrix}
 +
          2 \\
 +
          6 \\
 +
          4 \\
 +
          0
 +
        \end{bmatrix}\in W^\perp</math>
 
|}
 
|}
 
[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]
 
[[031_Review_Part_3|'''<u>Return to Sample Exam</u>''']]

Revision as of 08:57, 13 October 2017

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 W=\text{Span }\Bigg\{\begin{bmatrix} 2 \\ 0 \\ -1 \\ 0 \end{bmatrix},\begin{bmatrix} -3 \\ 1 \\ 0 \\ 0 \end{bmatrix}\Bigg\}.}   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} 2 \\ 6 \\ 4 \\ 0 \end{bmatrix}}   in  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 W^\perp?}   Explain.


Foundations:  
Recall that if  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 W}   is a subspace 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 \mathbb{R}^n,}   then
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 W^\perp=\{ \vec{v}\in \mathbb{R}^n ~: ~ \vec{v}\cdot \vec{w}=0 \text{ for all }w\in W\}.}


Solution:

Step 1:  
To determine whether the vector
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} 2 \\ 6 \\ 4 \\ 0 \end{bmatrix}}
is in  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 W^\perp,}   it suffices to see if this vector is orthogonal to
the basis elements 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 W.}
Notice that we have

       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{array}{rcl} \displaystyle{\begin{bmatrix} 2 \\ 6 \\ 4 \\ 0 \end{bmatrix}\cdot \begin{bmatrix} 2 \\ 0 \\ -1 \\ 0 \end{bmatrix}} & = & \displaystyle{2(2)+6(0)+4(-1)+0(0)}\\ &&\\ & = & \displaystyle{0.} \end{array}}

Step 2:  

       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{array}{rcl} \displaystyle{\begin{bmatrix} 2 \\ 6 \\ 4 \\ 0 \end{bmatrix}\cdot \begin{bmatrix} -3 \\ 1 \\ 0 \\ 0 \end{bmatrix}} & = & \displaystyle{2(-3)+6(1)+4(0)+0(0)}\\ &&\\ & = & \displaystyle{0.} \end{array}}

Hence, we conclude
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} 2 \\ 6 \\ 4 \\ 0 \end{bmatrix}\in W^\perp.}


Final Answer:  
        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} 2 \\ 6 \\ 4 \\ 0 \end{bmatrix}\in W^\perp}

Return to Sample Exam

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