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_{1}>e

_{1}+...+ <v,e

_{n}>e

_{n}. This appears to be widely used for many results (such as Gram-Schmidt), but the motivation for this is not clear to me. Not only that, I don't understand why this is the case (geometrically).

Obtaining the coefficients for a given vector (in terms of an orthonormal basis) using the inner product of the vector v with each basis element... why does this work?