The basic idea is this. Suppose that

is a

-space. You can completely specify a linear transformation from

to any other

-space

by demanding that

where

is a basis for

and the

are just any vectors in

(not necessarily different). How? Well, say you have made a choice about what the

go to, you still haven't defined a transformation on

itself. That said, if the map which takes

is to be a linear transformation you must take each

to

. Thus, if

is linear and satisfies

then the function must be defined by the rule

where

is the UNIQUE representation of

as a linear combination of the basis

. Conversely, you can check that the map defined that way does, in fact, satisfy the condition of being a linear transformation

with

. So, in your case you have that

and

For the second part, what would happen if

(where I put

to emphasize that it's the zero function)? What happens if you plug in

for

?