Use induction to prove that $T(a_1v_1+\dots+a_nv_n)=a_1T(v_1)+\dots+a_nT(v_n)$ ($T$ is a linear map)

Question

I don't know how to continue and get stuck with the inductive step.

Use induction to prove $$T(a_1v_1+\dots+a_nv_n)=a_1T(v_1)+\dots+a_nT(v_n)$$ where $T$ is a linear map.

Answer 1

Since $T$ is linear it's true for $n=2$, suppose now that it's true for $n=k$, then:

$T\left(a_{1}x_{1} + \ldots + a_{k}x_{k} + a_{k+1}x_{k+1}\right) = T\left(\left(a_{1}x_{1} + \ldots + a_{k}x_{k}\right) + a_{k+1}x_{k+1}\right) =^{*} T\left(a_{1}x_{1} + \ldots + a_{k}x_{k}\right) + T\left(a_{k+1}x_{k+1}\right) =^{**} T\left(a_{1}x_{1}\right) + \ldots + T\left(a_{k}x_{k}\right) + T\left(a_{k+1}x_{k+1}\right) = a_{1}T\left(x_{1}\right) + \ldots + a_{k}T\left(x_{k}\right) + a_{k+1}T\left(x_{k+1}\right)$

-* is true because the assumption for $n=2$;

-** is true because the assumption for $n=k$.