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**Unenlightened** **Let** $\displaystyle L:R^{3}\rightarrow R^{2}$ **be the linear mapping defined by **$\displaystyle L\left(x,y,z\right)=\left(x-y,y-z\right)$.

**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 $\displaystyle L = \left( \begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \end{array} \right)$. 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...