031 Review Part 2, Problem 1

Consider the matrix  $A={\begin{bmatrix}1&-4&9&-7\\-1&2&-4&1\\5&-6&10&7\end{bmatrix}}$  and assume that it is row equivalent to the matrix

$B={\begin{bmatrix}1&0&-1&5\\0&-2&5&-6\\0&0&0&0\end{bmatrix}}.$

(a) List rank  $A$  and  ${\text{dim Nul }}A.$

(b) Find bases for  ${\text{Col }}A$  and  ${\text{Nul }}A.$  Find an example of a nonzero vector that belongs to  ${\text{Col }}A,$  as well as an example of a nonzero vector that belongs to  ${\text{Nul }}A.$

Foundations:
1. For a matrix  $A,$  the rank of  $A$  is
${\text{rank }}A={\text{dim Col }}A.$
2.  ${\text{Col }}A$  is the vector space spanned by the columns of  $A.$
3.  ${\text{Nul }}A$  is the vector space containing all solutions to  $Ax=0.$

Solution:

(a)

Step 1:
From the matrix  $B,$  we see that  $A$  contains two pivots.
Therefore,
${\begin{array}{rcl}\displaystyle {{\text{rank }}A}&=&\displaystyle {{\text{dim Col }}A}\\&&\\&=&\displaystyle {2.}\end{array}}$

Step 2:
By the Rank Theorem, we have
${\begin{array}{rcl}\displaystyle {4}&=&\displaystyle {{\text{rank }}A+{\text{dim Nul }}A}\\&&\\&=&\displaystyle {2+{\text{dim Nul }}A.}\end{array}}$
Hence,  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 Nul }A=2.}

(b)

Step 1:
From 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 B,}   we see that  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}   contains pivots in Column 1 and 2.
So, to obtain a basis 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 \text{Col }A,}   we select the corresponding columns from  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.}
Hence, a basis 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 \text{Col }A}   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 \Bigg\{\begin{bmatrix} 1 \\ -1 \\ 5 \end{bmatrix}, \begin{bmatrix} -4 \\ 2 \\ -6 \end{bmatrix}\Bigg\}. }

Step 2:
To find a basis 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 \text{Nul }A} we translate the matrix equation  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 Bx=0}   back into a system of equations
and solve for the pivot variables.
Hence, 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{x_1-x_3+5x_4} & = & \displaystyle{0}\\ &&\\ \displaystyle{-2x_2+5x_3-6x_4} & = & \displaystyle{0.} \end{array}}
Solving for the pivot variables, 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{x_1} & = & \displaystyle{x_3-5x_4}\\ &&\\ \displaystyle{x_2} & = & \displaystyle{\frac{5}{2}x_3-3x_4.} \end{array}}
Hence, the solutions 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}   are of the form
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_3\begin{bmatrix} 1 \\ \frac{5}{2} \\ 1 \\ 0 \end{bmatrix}+x_4\begin{bmatrix} -5 \\ -3 \\ 0 \\ 1 \end{bmatrix}.}
Hence, a basis 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 \text{Nul }A}   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 \Bigg\{\begin{bmatrix} 1 \\ \frac{5}{2} \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -5 \\ -3 \\ 0 \\ 1 \end{bmatrix}\Bigg\}. }

Final Answer:
(a)     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{rank }A=2}   and 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 Nul }A=2.}
(b)     A basis 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 \text{Col }A}   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 \Bigg\{\begin{bmatrix} 1 \\ -1 \\ 5 \end{bmatrix}, \begin{bmatrix} -4 \\ 2 \\ -6 \end{bmatrix}\Bigg\}. }
and a basis 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 \text{Nul }A}   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 \Bigg\{\begin{bmatrix} 1 \\ \frac{5}{2} \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -5 \\ -3 \\ 0 \\ 1 \end{bmatrix}\Bigg\}. }

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