Linear Algebra

MATH 4450 - HOME WORK 5 (1) Let V be a R−vector space and < , > be an inner product. Prove that if {v1, · · · , vn} is a set of mutually orthogonal non-zero vectors, then this set is also linearly independent. (2) Let V be a vector space and < , > be an inner product. Then show that (a) < 0, v >= 0 for any v ∈ V . (b) Show that for a fixed u ∈ V , < u, v >= 0 for any v ∈ V , then u = 0. (c) Show that < v, w >= 0 for any w ∈ V implies that v = 0. (3) Let V be a vector space and < , > be an inner product and B = {v1, · · · , vn} be an orthogonal basis for V . If c1, · · · , cn ∈ K are scalars, then show that there is a unique v ∈ V such that < v, vi >= ci . (4) Use the Gram-Schmidt orthogonilisation method to the vectors (1, 0, 1), (1, 0, −1) and (0, 3, 4) to obtain an orthonormal basis for R 3 . (5) Let V be a vector space equipped with an inner product. Let W be subspace of V . Let W⊥ be its orthogonal complement. Define a map π : V → V as π(v) = v1 where v = v1 + v2 with v1 ∈ W and v2 ∈ W⊥. Show that (a) The map π is well-defined, linear with range W and that π 2 = π. The map π is called the projection map onto the subspace W. (b) Define ν := I − π. Show that ν is the projection map onto W⊥ i.e. show that ν is a linear map with range W⊥ and that ν 2 = ν. 1