Question:
Inner product and orthogonal vectors
1. Let V be an R-vector space with inner product ( - . - ).
(a) Let S = {b1, b2, ...} be a set of vectors in V. Define what it means for S to be an orthogonal set or an orthonormal set with respect to the inner product.
(b) Let V = R^4 and let ( - . - ) be the dot product. Apply the Gram-Schmidt orthoganlisation process to the set
{v1 = (1, 1, 1, 0), v2 = (-1, 0, 0, 2), v3 = (0, 0, 1, 1)}
Please give some intermediate steps and not just the final results.
(c) Let W be a subspace of V. What is the orthogonal complement of W in V?
(d) Let W = Span({v1, v2, v3}) with v1, v2, and v3 as in (b). What is the dimension of the orthogonal complement of W in V? Calculate.