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
${\text{rank }}A+{\text{dim Col }}A=n.$

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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