# Vector spaces v and w are finite dimensional prove that v×w is also finite dimensional

If two vector spaces v and w are finite dimensional,how do we prove that v×w is also finite dimensional.How do we express dim(v×w) in terms of dim(v) and dim(w)

• Its the sum of dimensions. – Wuestenfux Sep 12 at 7:53

Let $$\{v_1, v_2, ..., v_n\}$$ be a basis for $$V$$ (dimension $$n$$) and let $$\{w_1, w_2, ..., w_m\}$$ a basis for $$W$$ (dimension $$m$$).
It's not hard to show that $$\{(v_1, 0), (v_2, 0), ...(v_n,0), (0, w_1), (0, w_2), ..., (0, w_m) \}$$ is a basis for $$V \times W$$, since those vectors are linearly independent and you can write any vector in $$V \times W$$ as a linear combination of those.
That basis has $$n+m$$ elements, which makes $$dim(V \times W)=n+m$$.
Long story short, $$dim (V \times W) = dim(V) + dim(W)$$