(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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(vm) } 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.

3. 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).

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Another Linear Algebra proof about linear transformations

**Physics Forums | Science Articles, Homework Help, Discussion**