Let bv1vn be a basis of a subspace v of rnx1 let x be the

Question

Question 1:

Let B={v_1,...,v_n} be a basis of a subspace V of R^nx1. Let x be the nonzero vector x=a₁v₁+...+a_nv_n for scalars a_i. Let C={x, v₂,...,v_n}.

a) Show that if a₁ is not equal to 0 then C is also a basis.

b) Show that if a₁ =0 then C is not a basis.

Question 2:

Prove that if A is a real n by n matrix and if the expression x^TAy for x,y elements of R^nx1 does indeed define an inner product on R^nx1 then A must be symmetric and positive definite. Hint: consider <e_i| e_j>. Recall that a matrix A is positive definite if x^TAx>0 for every nonzero vector x.

Question 3:

Let V, W₁, W₂, and V₀ be subspaces of C^nx1. Let B={u₁, u₂,...,u_n} be an orthonormal basis of V. Define V₀^- = {u | u - v₀ for all v₀ that are elements of V₀}. Observe (DO NOT PROVE) that the intersection of W₁ and W₂, W₁ + W₂, and V₀^- are subspaces of V. By definition the sum W₁ + W₂ is said to be direct if the intersection of W₁ and W₂ consists only of the zero vector.

a) Suppose that {u₁,...,u_k} (where k is less than or equal to n ) is an orthonormal basis of V₀. Show that V₀^- = span{u_k+1,...,u_n}

And that

V₀ + V₀^- = V.

b) Show that the sum V₀ + V₀^-  is a direct sum.

c) Show that if W₁ + W₂ is a direct sum then every vector w in W₁ + W₂ has a unique representation as w= w₁+w₂ for w₁ in W₁ and w₂ in W₂.

d) Conclude that every vector v in V has a unique representation as v = v₀ + y with v₀ in V₀ and y in V₀^-.