Is change of basis a linear operation?

Question

If $T$ and $U$ are linear operators on $V$ and $\beta$ is a basis for $V$, is it true that $$[T + aU]_\beta = [T]_\beta + a[U]_\beta\ ?$$ I know that scalar multiplication is preserved, just not sure about the additive part.

Answer 1

I figured it out. Turns out it was pretty simple and I just had to write it out explicitly.

Let $v$ be a vector, $\beta = \{v_1,\ldots, v_n\}$, and $T(v) = \sum a_i v_i, U(v) = \sum b_i v_i$. We want to show that \begin{align*} [T + \lambda U]_\beta [v]_\beta &= ([T]_\beta + \lambda[U]_\beta)[v]_\beta \\ [T(v) + \lambda U(v)]_\beta &= [T(v)]_\beta + \lambda[U(v)]_\beta. \end{align*} The RHS is $[a_i] + \lambda [b_i]$, which is the same as the LHS: $[a_i + \lambda b_i]$.