Let be a basis for a subspace W of an inner product space V, and let z V. Prove

Question

Let be a basis for a subspace W of an inner product space V, and let z V. Prove that z "W perp" iff <z,v> = 0 for every v .

Note: "w perp" is a non empty subset of an inner product space V, and is the set of all vectors in V that are orthogonal to every vector in W. aka orthogonal complement

Explanation / Answer

Anyway, the first direction (=>) is easy, since ß is a subset of W. For the other direction, let w be an element of W. Then write w as a linear sum of elements of ß. Finally, use the assumption to show that = 0. Since this is true for all w in W, z must be in "W perp." hope that helped:)

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