Complete the following statements S v1 v2 vn is an orthonorm

Complete the following statements. S= {v_1, v_2,, v_n} is an orthonormal basis for an inner product space V iff Let W be a subspace of an inner product space V. Let S= {u_1, u_k} be an orthonormal basis for W. Then for any vector x V, the orthogonal projection of x on W is defined by Let W be a subspace of an inner product space V. Then a set U in V is the orthogonal complement to W iff Let A R^m times n. Then the null space of A, denoted by N(A), is defined by

Solution

(a) S = { v₁ , v₂ , ...v_n } is an orthonormal basis for an inner product space V if

i) v_i .v_j = 0 for all i , j (1 i , j n) ;

ii) every element of V can be expressed as a linear combination of v₁ , v₂ , ..., v_n;

iii) the vectors v₁ , v₂ , ..., v_n are linearly independent; and

iv) the magnitude of each v_i is 1 i.e each v_i is a unit vector.

(b) The orthogonal projection of x on W is defined by proj_v1 x + proj_v2 x + ...+ proj_vn x . Also proj_u v is defined by proj_u v = [< v, u >/ < u, u>} u.

(c) The set U is the orthogonal complement of W iff u_i . w_j = 0 for all u_i U and all w_j W.

(d) The null space of a mx n matrix A, denoted N( A) , is the set of all solutions to the homogeneous equation Ax = 0. Written in set notation, we have N( A ) = {x : x Rⁿ and Ax = 0 }