Image of L is contained in M

Let A be a 4x3 matrix and let B be a 4x4 matrix. BA is then a 4x3 matrix. Let L be a linear transformation from R^4 to R^3 defined by L(x) = xA where x is a vector. Let M be the linear transformation from R^4 to R^3 defined by M(x) = xBA.

Show that the image of M is contained in the image of L.

What I have-

Take any vector x. The image of L and M will be the span of the rows of A and the rows of BA, or the set of all linear combinations. So, we have:

[x1 x2 x3 x4]*A = x1A1 + x2A2 + x3A3 + x4A4

and

[x1 x2 x3 x4]*BA = x1BA1 + x2BA2 + x3BA3 + x4BA4

I'm stuck at this point. I think I have to invoke the linearity fact and add these together somehow but i'm not seeing it…

Thanks.

Re: Image of L is contained in M

It is more simple: you needn't the concept of linear combination, only the definition of image. If $\displaystyle y\in\mbox{Im }M,$ then there exists $\displaystyle x\in\mathbb{R}^4$ such that $\displaystyle y=xBA=(xB)A.$ As a consequence, $\displaystyle y\in\mbox{Im }L.$