Problem Statement:
Let $T:V\rightarrow V$ be a linear transformation. The following are equivalent:
($i$)
$\exists \mathcal{B}=\{v_1,...,v_n\}\subset V$ such that $$[T]_\mathcal{B}=\begin{pmatrix} \lambda_1 & 0 & \dots & 0 \\ 0 & \lambda_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \lambda_n \end{pmatrix}$$
($ii$)
$\exists u_1,...,u_k\in \mathbb{F}$ and subspaces $V_1,...,V_k$ of a Vector Space $V$ such that $V=\bigoplus^{k}_{1}V_i$, and $\forall i\in \{1,...,k\}$ the action of $T$ on $V_i$ is given by multiplication by $u_i$.
I only need clarification. My Problem is that I do not understand what "the action of $T$ on $V_i$ is given by multiplication by $u_i$" means. Does it mean that if $v_i\in V_i$, then $T(v_i)=u_iv_i$?
My professor did not define what it means and it doesn't seem absolutely clear from the context.
Oh, and $[T]_\mathcal{B}$ is the matrix representation of the linear transformation $T$