Let

$211555$

$132$ A$=5105$

$($b$)$Explain why B $=\{$v$1,$v$2,$v$3\}$ is a basis for R$3$ and give the change of basis matrix $[$x$]$B$,$ that is find the matrix C so that

$$c$1$

$[$x$]$B $=$ C$$x $=$c$2$ where $$x $=$ c$1$v$1+$ c$2$v$2+$ c$3$v$3$

c$3$

$($c$)$Give matrices P and D so that A $=$ P DP $1$ and D is a diagonal matrix.

$($d$)$For a vector $$u whose entries sum to $1,$ approximate Ak$$u for large k$.$