For each linear transformation below: (a) Find the matrix representation of $T$, (b) Calculate $n(T)$, (c) Calculate $r(T)$, (d) Graph the image in either $\real{2}$ or $\real{3}$ as appropriate, (e) How many dimensions are lost?, and (f) How many dimensions are preserved?

1. $\ltdefn{T}{\complex{3}}{\complex{3}}$ given by $\lt{T}{\colvector{x\\y\\z}} = \colvector{x\\x\\x}$
2. $\ltdefn{T}{\complex{3}}{\complex{3}}$ given by $\lt{T}{\colvector{x\\y\\z}} = \colvector{x\\y\\0}$
3. $\ltdefn{T}{\complex{3}}{\complex{2}}$ given by $\lt{T}{\colvector{x\\y\\z}} = \colvector{x\\x}$
4. $\ltdefn{T}{\complex{3}}{\complex{2}}$ given by $\lt{T}{\colvector{x\\y\\z}} = \colvector{x\\y}$
5. $\ltdefn{T}{\complex{2}}{\complex{3}}$ given by $\lt{T}{\colvector{x\\y}} = \colvector{x\\y\\0}$
6. $\ltdefn{T}{\complex{2}}{\complex{3}}$ given by $\lt{T}{\colvector{x\\y}} = \colvector{x\\y\\x+y}$