Given two $3-$dimensional vectors a$,$ b$,$ and three matrices A in R$23,$ B in R$32,$ C in R$23,$ scalars

$1,$ $2$ with the values below:

a $=$

$$

$$

$1$

$2$

$3$

$$

$$ b $=$

$$

$$

$4$

$5$

$6$

$$

$$ A $=$

$123$

$456$

B $=$

$$

$$

$78$

$910$

$1112$

$$

$$ C $=$

$100$

$010$

$1=2,$ $2=3$

$($a$)$ Compute the following operations by hand and show your work: $(15$ points$)$

$($i$)$ Vector operations: a $+$ b$,$ $1$a$,$ a $$ b$,$ $1$a $+$ $2$b where $$ denotes component$-$wise

multiplication. $(3$ points$)$

$($ii$)$ Matrix operations: $1$A$,$ A $+$ B$,$ A $+$ C$.(3$ points$)$

$($iii$)$ Transpose operations: $($AB$),$ B$$A$,($AT $)$T $,($A $+$ C$).(3$ points$)$

$($iv$)$ Inner products and outer product: $$a$,$ b$,$b$,$ a$,$a$,$ a$,$b$,$ b$,$ $1$a$,$ b$,$$1$a$,$ b$,$ ba$.$