- #1

nietzsche

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## Homework Statement

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.

## Homework Equations

## 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}.