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)

  • $\begingroup$ Its the sum of dimensions. $\endgroup$ – 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, $dim (V \times W) = dim(V) + dim(W)$


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