Linear algebra change of coordinates

[sweetiepi]

Homework Statement

Consider a transformation T in the vector space consisting of linear transformation T: V -> V with a basis B = {b1, ..., bn} for V. Show that {k1, ..., kv} is a basis for Ker(T) if and only if {[k1]B, ..., [kv)B} is a basis for Nul([T]B) and that {r1, ..., rq} is a basis for Range(T) if and only if {[r1]B, ..., [rq]} is a basis for Col([T]B).

The notation [ ]B indicates that it is a coordinate vector.

Homework Equations

Not sure that there are any.

The Attempt at a Solution

I just don't get how to relate everything... I don't know how to think about it all which means I don't really have any good attempts.

 

[kurt.physics]

I understand the idea of a basis, but I don't know how to apply it here. Could someone explain it to me?