$\dim(\mathbb{V}_1 \oplus ...\oplus \mathbb{V}_k) = \dim\mathbb{V}_1+...+\dim\mathbb{V}_k$

Question

Problem:

If $\mathbb{X}$ is a vector space and $\mathbb{V}_1 \oplus ...\oplus \mathbb{V}_k$ ($k \leq n$) a direct sum of subspaces of $\mathbb{X}$. Show that

$$\dim(\mathbb{V}_1 \oplus ...\oplus \mathbb{V}_k) = \dim\mathbb{V}_1+...+\dim\mathbb{V}_k\leq{n}$$

Some hint, please.

Answer 1

We have (for subspaces $\mathbb V_1$ and $\mathbb V_2$ of $\mathbb X$) $$\dim(\mathbb V_1\oplus\mathbb V_2)=\dim(\mathbb V_1)+\dim(\mathbb V_2) -\dim(\mathbb V_1\cap\mathbb V_2). $$

Prove this, then you can generalize to $k$-many subspaces. Hope I helped!