6. Suppose V is a real inner product space. (a) Show that if u, v € V with ||u|| = ||v||, then u + v is orthogonal to u – v. (b) Provide an example of a real inner product space V and u, v Є V such that u + v is not orthogonal to u - v.

11. Let (V, (, )) be a complex inner product space. For vectors v, w, set (v, w) R = ¹/[(v, w)+(w, v)]. Consider V to be a real vector space. Is (V, (, )R) an inner product space? Support your answer with a proof.

Show that if X and Y Hausdorff spaces, then so is the product space X × Y. are

Consider the space P₂ which is the vector space of polynomials of degree not greater than two. We define the inner product in P2 according to the formula -Lpqdz. Also, consider the following two vectors (a) Calculate the inner product (p, q) (b) Calculate the distance between vectors p and q. (c) Calculate the norm ||q||, of vector q. (p, q) (e) For vectors p and q, calculate the inner product (p - q, p + 2q). p = x² = 5x + 1, q= −2x²+x. (d) For vectors p and q, find the orthogonal projection of p on q (projp). Your answer must be a vector from P2, i.e. a polynomial of degree 2 or less.

3) Let V = {(5), a, b e R*} note R* = {r € R]r > 0} ). Define “addition" on V by G) +C)=G) Further, define "scalar multiplication for ce R by c () = (). Prove that V is a (b,b2 vector space over R by showing that the set is closed under all the scalar multiplication and vectors addition and that all the axioms pertaining to these operations hold.

Let V be an inner product space with dim V = 2, and suppose that L = {u+v, u+2v} is a linearly independent subset of V. Prove that if 1 is orthogonal to both u and v, then 1 = assume that V contains tuples. Also, be sure to use only results from our linear algebra course in your proof.) 0. (Note: You Imay not