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

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.