**Let** **be the linear mapping defined by ** .

**Find the standard 2x3 matrix representation of L and compute L(1,1,1).**

Show that the kernel of L is a subspace of R^{3}. What is the dimension of the kernel?

State the Dimension Theorem. Hence or otherwise find the image of R^{3} under the transformation L.

So I have

. L(1,1,1)=(0,0).

The kernel of L has basis (1,1,1) and has dimension 1, and I've proven it is a subspace of R^{3}.

My problem is with the final piece of the question.

The Dimension Theorem states that given a vector space, any two bases have the same cardinality...

How am I supposed to use that to find the image of R^{3} under L?

Many thanks...