031 Review Part 3, Problem 11

Suppose  $\{{\vec {u}},{\vec {v}}\}$  is a basis of the eigenspace corresponding to the eigenvalue 0 of a  $5\times 5$  matrix  $A.$

(a) Is  ${\vec {w}}={\vec {u}}-2{\vec {v}}$  an eigenvector of  $A?$  If so, find the corresponding eigenvalue.

If not, explain why.

(b) Find the dimension of  ${\text{Col }}A.$

Foundations:
1. An eigenvector  ${\vec {x}}$  of a matrix  $A$  corresponding to the eigenvalue  $\lambda$  is a nonzero vector such that
$A{\vec {x}}=\lambda {\vec {x}}.$
2. By the Rank Theorem, if  $A$  is a  $m\times n$  matrix, then
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\text{rank }}A+{\text{dim Col }}A=n.}

Solution:

(a)

Step 1:
First, notice
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {w}}\neq {\vec {0}}}
since  $\{{\vec {u}},{\vec {v}}\}$  is a basis of the eigenspace corresponding to the eigenvalue 0 of  $A.$
Also, we have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A{\vec {u}}={\vec {0}}}   and  $A{\vec {v}}={\vec {0}}$
since  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {u}}}   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 \vec{v}}   are eigenvectors 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 A}   corresponding to the eigenvalue 0.

Step 2:
Now, 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{A\vec{w}} & = & \displaystyle{A(\vec{u}-2\vec{v})}\\ &&\\ & = & \displaystyle{A\vec{u}-2A\vec{v}}\\ &&\\ & = & \displaystyle{\vec{0}-2\cdot \vec{0}}\\ &&\\ & = & \displaystyle{\vec{0}.} \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 \vec{w}}   is an eigenvector 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 A}   corresponding to the eigenvalue  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 0.}

(b)

Step 1:
Since  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},\vec{v}\}}   is a basis for the eigenspace 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 A}   corresponding to the eigenvalue 0, we know 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 \text{dim Nul }A=2.}

Step 2:
Then, by the Rank Theorem, 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{5} & = & \displaystyle{\text{dim Col }A+\text{dim Nul }A}\\ &&\\ & = & \displaystyle{\text{dim Col }A+2.} \end{array}}
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 \text{dim Col }A=3.}

Final Answer:
(a)     See solution above.
(b)     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=3}

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