$32.$ Let T: $33$ be a linear operator and

B $=\{$v$1,$ v$2,$ v$3\}$ a basis for $3.$ Suppose

T $($v$1)=$

$1$

$2$

$$

$1$

$$ T $($v$2)=$

$5$

$0$

$0$

T $($v$3)=$

$1$

$1$

$$

$2$

$$

a$.$ Determine whether

w $=$

$1$

$2$

$2$

$$

$$

$$

$$

is in the range of T$.$

b$.$ Find a basis for R$($T $).$

c$.$ Find dim$($N $($T $)).$