Suppose that V and W are real vector spaces and T in $($V$,$W$).$ Define

Tc $$ Vc $->$Wc by

$(+)=+$

for all $,$ in $.$

$($a$)$ Show that $$ is a $($complex$)$ linear map from $$ to $.$

$($b$)$ Show that $$ is injective if and only if $$ is injective.

$($c$)$ Show that range $=$ if and only if range $=$