Euclidean vector space is direct sum of subspace and orthogonal complement (theorem & proof & notes)

A Euclidean vector space is the direct sum of a subspace and the orthogonal complement of this subspace. I explain orthogonal complements and prove this theorem!
orthogonal complement (0:21)
theorem & proof (3:20)
notes (12:16)

*I am sorry for any incorrect pronunciation of English words*

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[xanderlewis]

Your definition of the orthogonal complement of a subspace U is interesting... I’ve only ever seen it defined as the set of vectors orthogonal to ALL vectors in U, rather than just to each vector in a basis for U. Of course, because of the bilinearity of the inner product, they are equivalent anyway.