Matrix representation of linear transformation

[toni07]

Let V and W be two finite-dimensional vector spaces over the field F. Let B be a basis of V, and let C be a basis of W. For any v 2 V write [v]B for the coordinate vector of v with respect to B, and similarly [w]C for w in W. Let T : V -> W be a linear map, and write [T]C B for the matrix representing T with respect to B and C.
a) Prove that a vector v in V is in the kernel of the linear map T if and only if the vector [v]B in F^dimV is in the nullspace of the matrix [T]C B.
b) Prove that a vector w in W is in the range of the linear map T if and only if the vector [w]C in F^dimW is in the column space of the matrix [T]C B.
I don't know how to go about this question, any guidelines would really help.

 

[HallsofIvy]

Given an ordered basis, B, for V and an ordered basis, C, for W. We write the matrix representation for T: V-> W by applying T to each vector in B, in turn, then writing the result as a linear combination of the vectors in C. The coefficients form the columns of [T]C B. That is because the first vector in B would be represented by $\begin{bmatrix}1 \\ 0 \\ 0 \\ ... \\ 0\end{bmatrix}$, the second by $\begin{bmatrix}0 \\ 1 \\ 0 \\ ... \\ 0\end{bmatrix}$, etc. Then
$$\begin{bmatrix}a & b & c \\ d & e & f\\ h & i & j\end{bmatrix}\begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}= \begin{bmatrix}a \\ d \\ h\end{bmatrix}$$
That is, multiplying the matrix representation of T by the matrix representation of each basis vector in B gives a column of the matrix, the matrix representation of the corresponding basis vector in C.

A vector, v, is in the kernel of T if and only if Tv= 0 which would be written, in terms of basis C, as the number 0 times each vector in C. That would be represented by the matrix $\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ so that the representation of v in B is in the nullspace of [T]C B.

Similarly, since the columns of [T]C B give the coefficients of the linear combinations of basis vectors in C, there is a one to one correspondence between the range of T and the linear combinations of the columns.