Let V and W be two nite-dimensional vector spaces over the eld 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.
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.