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## Homework Statement

Given:

T is a linear transformation from V -> W and the dim(V) = n and dim(W) = m

Prove:

If β = {v1, ..., vm} is a basis of V, then { T(v1), ..., T(v

**m**) } spans the image of T.

**NOTE:**because of bad hand writing I can't tell if the bold is suppose to be an 'm' or an 'n'. I think 'm' because that makes more sense to me.

## The Attempt at a Solution

Let A = { T(v1), ..., T(vm) } .

We must show that the vectors in A are L.I., and that the dim(A) is m.

If we show the vectors of A are L.I. then since there are m vectors we know the dimension is m.

Then there must be only the trivial solution to c1T(v1) + ... + cmT(vm) = 0 .

Or, by linearity, T(c1v1 + ... + cmvm) = 0 .

*Note: I don't think I can jump straight to the next step. (in a similar thread of mine I did so by utilizing the fact that the null space is 0).*

Since the vectors v1...vm form a basis, they are L.I. and only the trivial solution exists.

Therefore span(A) = Im(T).