How to show that matrices maintain linearity Is it enough to show that the rules of matrices hold with adittion and scalar multiplication to prove that the map that sends a linear transformation to its matrix representation is a linear transformation itself? If not, can I see how it is shown then? Thanks
 A: Yes, assuming we have a finitely dimensional vector spaces S and W, we know there is a basis of these vector spaces. Because of this, the linear transformation T is uniquely determined by the coefficients of the linear combination which is expressed in a transformation matrix with respect to the basis $\alpha$ for S and $\beta$ for W, that is:
$$\textbf{T} : \textbf{S} \rightarrow \textbf{W} $$
Then $\vec{x}$ is some vector in $\textbf{S}$ so we have:
$$\vec{x} = a_1\alpha_1 + \cdots + a_n\alpha_n$$
And the linear transformation takes vectors from S to W then the mapped vector $\vec{y}$ in W can be expressed in terms of the basis vectors $\beta$:
$$\vec{y} = b_1\beta_1 + \cdots + b_k\beta_k$$
Then the linear transformation is:
$$T(\vec{x}) = T(a_1\alpha_1 + \cdots +a_n\alpha_n)$$
And by linearity we have:
$$ = a_1T(\alpha_1) + \cdots + a_nT(\alpha_n)$$
Then by the basis $\beta$ we have:
$$ = a_1(b_{11}\beta_1 + \cdots + b_{1k}\beta_k) + \cdots + a_n(b_{k1}\beta_1 + \cdots + b_{kk}\beta_k)$$
And these coefficients are the entries in the transformation matrix with respect to $\alpha$ and $\beta$
