This might be a rather stupid question. I have read that the Union of two subspaces is a subspace iff one of the subspaces is contained in the other. However, we also know that W and it's orthogonal complement span the entrie vector space. Since span is a subspace, how does this reconcile with the above statement? W and it's orthogonal complement are disjoint except zero vector.
2026-03-27 13:38:12.1774618692
Spanning by W and its Orthogonal Complement
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Consider an example. Take V = $\mathbb{R}^2$. Let W = the x-axis and U = W$^\perp$ = the y-axis. Think of the vector (1,1). It is NOT in W nor is it in U. The union of two vector spaces in general does not contain the sum of vectors w + u, where w $\in$ W and u $\in$ U. That is why we need the sum of W and W$^{\perp}$ to equal V.