Prove about direct sum

Question

Prove if $W=W_1+\cdots+W_k$ then $\dim(W)=\dim(W_1)+\cdots+\dim(W_k)$ I'm trying to prove this but is a little complicated for me. Can someone help me?

Answer 1

Suppose that for $\;1\le i\neq j\le n\;$ , we have

$$x\in\ker (\lambda_iI-T)\cap\ker(\lambda_jI-T)\implies Tx=\lambda_ix=\lambda_jx$$

and since $\;\lambda_i\neq\lambda_j\;$ , we get that $\;x=0\;$ .

Generalizing over the above, you get

$$\;\sum_{i=1}^k W_i=\bigoplus_{i=1}^k W_i = \text{ direct sum}\;$$