Linear Transformations: Explaining the Theorem

[finkeljo]

I don't quite understand the idea that (as my book says) every linear transformation with domain Rn and codomain Rm is a matrix transofrmation... I mean i get the idea of what a linear transformation is (sorta like a function) but it gives the theorem:

Let T: Rn -> Rm be linear. Then there is a unique m x n matrix

A=[T(e1)T(e2)...T(en)]

Can some one just explain that a little bit? It may seem simple but I don't think my book does a good job providing enough background for the theorems they state.

 

[Fredrik]

Can some one just explain that a little bit?

See this post. Ask if there's something you don't understand.

 

[HallsofIvy]

Note that
$$\begin{bmatrix}a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{23} & ... & a_{2n} \\ ... & ... & ... & ... \\ a_{m1}& a_{m2} & ... & a_{mn}\end{bmatrix}\begin{bmatrix}1 \\ 0 \\ ... \\ 0\end{bmatrix}= \begin{bmatrix}a_{11} \\ a_{21} \\ ... \\a_{m1}\end{bmatrix}$$

What do you get if you multiply that same matrix by
$$\begin{bmatrix} 0 & 1 & ... & 0\end{bmatrix}$$
etc.?

Do you see that applying any linear transformation to the basis vectors in[math]R^n[math] gives you the columns of the matrix representation?

(This is, by the way, "unique" only in the standard bases for $R^n$ and $R^m$. If L is a linear transformation from vector space U to vector space V, you can get different matrix representations for every different choice of basis for U or V.)