Consider the inner product R R defined by (x, y) = [x] [12] [v]. (i) Is...

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Consider the inner product R R defined by (x, y) = [x] [12] [v]. (i) Is it possible to find a vector u R such that (u, u) =-1? Give a reason. (ii) Verify that {(1,0), (1, 1)} is an orthonormal set with respect to this inner product. (iii) Let z E R. Determine the scalars a, a2 R in terms of the inner product, if z = a(1,0) + a(-1,1). (b) Consider the plane U spanned by the vectors (1,1,0) and (0,1,1). Using the Gram- Schmidt procedure, find an orthonormal basis for U with respect to the dot product. Consider the inner product R R defined by (x, y) = [x] [12] [v]. (i) Is it possible to find a vector u R such that (u, u) =-1? Give a reason. (ii) Verify that {(1,0), (1, 1)} is an orthonormal set with respect to this inner product. (iii) Let z E R. Determine the scalars a, a2 R in terms of the inner product, if z = a(1,0) + a(-1,1). (b) Consider the plane U spanned by the vectors (1,1,0) and (0,1,1). Using the Gram- Schmidt procedure, find an orthonormal basis for U with respect to the dot product.

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