T: V --> W is a linear transformation where V and W are finite dimensional.
If dim V is less than or equal to dim W, then T is one-to-one. True or false?
The Attempt at a Solution
First of all, I'm assuming that im(T) = W. Is that correct? If so, dim(im T) = dim(W).
Dimension thm says dim(V) = dim(ker T) + dim(im T).
So if dim(V) is less than or equal to dim(im T), then dim(ker T) = 0. Which means ker T is is as small as it can be, so T should be one-to-one.
But this is wrong. The answer is supposed to be false. Can anyone help?