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3. guided proof prove that if w is orthogonal to...

# Question: guided proof prove that if w is orthogonal to...

###### Question details

Guided Proof :

Prove that if w is orthogonal to each vector in S = {v1, v2, . . . , vn}, then w is orthogonal to every linear combination of vectors in S. Getting Started: To prove that w is orthogonal to every linear combination of vectors in S, you need to show that their inner product is 0.

(i) Write v as a linear combination of vectors, with arbitrary scalars c1, . . . , cn, in S.

(ii) Form the inner product of w and v.

(iii) Use the properties of inner products to rewrite the inner product _w, v_ as a linear combination of the inner products _w, vi_, i = 1, . . . , n.

(iv) Use the fact that w is orthogonal to each vector in S to lead to the conclusion that w is orthogonal to v.