Linear Algebra | Part IB, 2004

Let VV be a real nn-dimensional inner-product space and let WVW \subset V be a kk dimensional subspace. Let e1,,ek\mathbf{e}_{1}, \ldots, \mathbf{e}_{k} be an orthonormal basis for WW. In terms of this basis, give a formula for the orthogonal projection π:VW\pi: V \rightarrow W.

Let vVv \in V. Prove that πv\pi v is the closest point in WW to vv.

[You may assume that the sequence e1,,ek\mathbf{e}_{1}, \ldots, \mathbf{e}_{k} can be extended to an orthonormal basis e1,,en\mathbf{e}_{1}, \ldots, \mathbf{e}_{n} of VV.]

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