How to prove that the sum of matrices of two linear maps, $Mμ+Mv$, equals to the matrix of sum of two linear maps $M(μ+v)$？

Question

I have a problem that suppose $μ,v$ are two linear maps belongs to $Hom(K^n,K^m)$

And $M_μ, M_v$ are the matrices of these two linear maps and $M_{(μ+v)}$ is the matrix of the sum of two linear maps.

So, how to prove that $M_μ + M_v = M{_(μ+v)}$?

I don't know how express the matrix of the sum of two linear maps so I get stuck. Can anyone help me out? Thank you!

Picture below is the definition of the matrix of a linear map, hope it can help

Answer 1

Start by showing the following:

• For any transformation $T$ as in your definition and $x = [x_1\;x_2\;\cdots\;x_n] \in K^n$, $[T]_\beta^\gamma x = T\left[\sum_i x_iv_i \right]$

• For any two matrices $M_1,M_2 \in K^{m \times n}$ are the same if and only if $M_1x = M_2x$ for every vector $x \in K^n$.

From there it suffices to note that for any $x = [x_1\;x_2\;\cdots\;x_n]$, we have $$ M_{(\mu + \nu)} x = (\mu + \nu)\left[\sum x_i v_i \right] = \mu\left[\sum x_i v_i \right] + \nu\left[\sum x_i v_i \right] = M_\mu x + M_\nu x = (M_\mu + M_\nu)x $$ Since this is true for all $x$, it must be that $M_{(\mu + \nu)} = M_\mu + M_\nu$.