Let $T$ be a linear operator over $V$ with dim$(V)=n$ and let the ordered set $B=${$v_1,v_2,...v_n$} be a basis for $V$. Furthermore, let $A=[T]_B$ be block-diagonal matrix (that is $$A=\bigoplus_{i=1}^{k} A_i$$ with block sizes $m_1,m_2,...m_k$ in that order). Then the subspaces (of $V$) $V_1,...V_k$ defined by $V_i=$span$(v_{f_i},v_{f_i+1},...,v_{m_i})=$span$(B_i)$ where $$f_i=1+ \sum_{j=1}^{i-1}m_j$$ for $2\le i\le k$ and $f_1=1$ are $T$-invariant and $$V=\bigoplus_{i=1}^{k}V_i$$ where $$[T_{V_i}]_{B_i}=A_i$$
My attempt: $B_i=${$v_{f_i},v_{f_i+1},...,v_{m_i}$} is a basis of $V_i$, also $\bigcup_{i=1}^{k}B_i=B$ then this implies that $V=\bigoplus_{i=1}^{k}V_i$
But I don´t know how to prove that $V_i$ is $T$-invariant for $1\le i\le k$ and that $[T_{V_i}]_{B_i}=A_i$( I know that I need to show that $T(V_i)\subseteq V_i$ )
I would really appreciate If you can help me with this problem:)