Show that taking the coordinates of a vector WRT a basis is a linear transformation

nietzsche

1. The problem statement, all variables and given/known data

Suppose U is a finite dimensional vector space and A = {u1, u2, ... , un} is a basis of U. Define T : U -> R(nx1) by T(v) = [v]_A.

(In other words: U is an n-dimensional vector space, A is a basis for U, and T is the transformation that takes a vector in U and finds its coordinate vector with respect to the basis A.)

Show that T is a linear transformation. Find kerT.


2. Relevant equations



3. The attempt at a solution

If v and w are arbitrary vectors in U and a and b are scalars, we have

T(av+bw)
= [av+bw]_A
=[av1+bw1, ... , avn+bwn]_A
=[av1 + ... + avn]_A + [bw1 + ... + bwn]_A
=a[v1 + ... + vn]_A + b[w1 + ... + wn]_A
=aT(v) + bT(w)

So T is a linear transformation.

I just don't see if what I'm doing here is justified. Please help! Thanks.


And for finding kerT, since A is a basis, it is linearly independent, so the only way to get the zero vector is to multiply everything by 0, so kerT = {0}.