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

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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"> Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?
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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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    \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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::<math>\begin{bmatrix}
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          1  \\
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          0 \\
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          2
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        \end{bmatrix},
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        \begin{bmatrix}
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          3  \\
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          1 \\
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          1
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        \end{bmatrix},
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        \begin{bmatrix}
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          -2  \\
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          -1 \\
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          1
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        \end{bmatrix},
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        \begin{bmatrix}
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          5  \\
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          2 \\
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          2
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        \end{bmatrix}</math>
  
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Foundations: &nbsp;  
 
!Foundations: &nbsp;  
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|-
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|'''1.''' &nbsp;<math style="vertical-align: -1px">\text{dim Col }A</math>&nbsp; is the number of pivots in &nbsp;<math style="vertical-align: 0px">A.</math>
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|-
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|'''2.''' A set of vectors &nbsp;<math style="vertical-align: -4px">\{\vec{v_1},\vec{v_2},\ldots,\vec{v_n}\}</math>&nbsp; is linearly independent if
 
|-
 
|-
 
|
 
|
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::the only solution to &nbsp;<math style="vertical-align: -4px">x_1\vec{v_1}+x_2\vec{v_2}+\cdots+x_n\vec{v_n}=\vec{0}</math>&nbsp; is the trivial solution.
 
|}
 
|}
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
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|-
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|We begin by putting these vectors together in a matrix. So, we have
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|-
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|
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::<math>   
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    \begin{bmatrix}
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          1 & 3 & -2 & 5 \\
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          0 & 1  & -1 & 2 \\
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          2 & 1 & 1 & 2
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        \end{bmatrix}.</math>
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|-
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|Now, we row reduce this matrix. We get
 
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|-
 
|
 
|
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&nbsp; &nbsp; &nbsp; &nbsp;<math>\begin{array}{rcl}
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\displaystyle{\left[\begin{array}{cccc} 
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          1 & 3 & -2 & 5 \\
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          0 & 1  & -1 & 2 \\
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          2 & 1 & 1 & 2
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        \end{array}\right]} & \sim & \displaystyle{\left[\begin{array}{cccc} 
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          1 & 3 & -2 & 5 \\
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          0 & 1  & -1 & 2 \\
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          0 & -5 & 5 & -8
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        \end{array}\right]}\\
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&&\\
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& \sim & \displaystyle{\left[\begin{array}{cccc} 
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          1 & 3 & -2 & 5 \\
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          0 & 1  & -1 & 2 \\
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          0 & 0 & 0 & 2
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        \end{array}\right]}
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\end{array}</math>
 
|}
 
|}
  
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|
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|Now, we have 3 pivots in this matrix. So, the dimension of the column space of the matrix we started with is 3.
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|-
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|Hence, the dimension of the subspace spanned by these vectors is &nbsp;<math style="vertical-align: 0px">3.</math>
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|-
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|When we row reduced the matrix, we had a column that did not contain a pivot.
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|-
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|This means we have a free variable in the system corresponding to &nbsp;<math style="vertical-align: 0px">Ax=0.</math>&nbsp;
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|-
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|So, these vectors are not linearly independent.
 
|}
 
|}
  
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!Final Answer: &nbsp;  
 
!Final Answer: &nbsp;  
 
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|-
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|&nbsp;&nbsp; &nbsp; &nbsp;   
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|&nbsp;&nbsp; &nbsp; &nbsp;  The dimension is &nbsp;<math style="vertical-align: 0px">3</math>&nbsp; and the vectors are not linearly independent.
 
|}
 
|}
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[[031_Review_Part_2|'''<u>Return to Sample Exam</u>''']]
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[[031_Review_Part_2|'''<u>Return to Review Problems</u>''']]

Latest revision as of 13:13, 15 October 2017

Find the dimension of the subspace spanned by the given vectors. Are these vectors linearly independent?

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 \\ 0 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -2 \\ -1 \\ 1 \end{bmatrix}, \begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix}}
Foundations:  
1.  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 \text{dim Col }A}   is the number of pivots 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 A.}
2. A set of vectors  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{{\vec {v_{1}}},{\vec {v_{2}}},\ldots ,{\vec {v_{n}}}\}}   is linearly independent if
the only solution to  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 x_1\vec{v_1}+x_2\vec{v_2}+\cdots+x_n\vec{v_n}=\vec{0}}   is the trivial solution.


Solution:

Step 1:  
We begin by putting these vectors together in a matrix. So, 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{bmatrix} 1 & 3 & -2 & 5 \\ 0 & 1 & -1 & 2 \\ 2 & 1 & 1 & 2 \end{bmatrix}.}
Now, we row reduce this matrix. We get

       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{\left[\begin{array}{cccc} 1 & 3 & -2 & 5 \\ 0 & 1 & -1 & 2 \\ 2 & 1 & 1 & 2 \end{array}\right]} & \sim & \displaystyle{\left[\begin{array}{cccc} 1 & 3 & -2 & 5 \\ 0 & 1 & -1 & 2 \\ 0 & -5 & 5 & -8 \end{array}\right]}\\ &&\\ & \sim & \displaystyle{\left[\begin{array}{cccc} 1 & 3 & -2 & 5 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 2 \end{array}\right]} \end{array}}

Step 2:  
Now, we have 3 pivots in this matrix. So, the dimension of the column space of the matrix we started with is 3.
Hence, the dimension of the subspace spanned by these vectors 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 3.}
When we row reduced the matrix, we had a column that did not contain a pivot.
This means we have a free variable in the system corresponding to  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 Ax=0.}  
So, these vectors are not linearly independent.


Final Answer:  
       The dimension 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 3}   and the vectors are not linearly independent.

Return to Review Problems

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