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 + vis not orthogonal to u - v.

6)(a)Proof given below. (b)For a real inner product space V = R2 Let u = ( 1 , 1 ) and v = ( 1 , 0…

Answered: 6. Suppose V is a real inner product…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Assume that V is a finite dimensional real inner product space and T = L(V) and self- adjoint. Show that T²+bT+cI is invertible if b² < 4c. Additionally, provide an example of a real inner product space V and T € L(V) and real numbers b, c with b² < 4c such that T² +bT+cI is not invertible. (Hint: The example shows that the hypothesis that T is self-adjoint is necessary, even for real vector spaces.)
Let V be a finite dimensional inner product space and T a non negative operator on V. Then T has a unique non negative square root, that is, there is one and only one non negative operator N on V such that N^2=T.
Give example on the inner product and same time is normed space
Let p(x) = a, +a,x+ a,x* +a,x' and g(x) = 6, + 6x+ 6,x + b,x' be vectors in P, with the standard inner product space on P,defined by = a,b, + a,b, + a,b, + a,b;. a) Find the distance between fand g f - | angle of f(x) and h(x) b) Use the Gram-Schmidt process to form an orthogonal set for the functions f(x) = 1-x-x* + 2x', g(x) = -2+ 2x - x and h(x) = x - x* + 2x
22. Show that there do not exist scalars c₁, c₂, and c² such that c₁(1,0, 1,0) + c₂(1, 0, −2, 1) + c3(2, 0, 1, 2) = (1, -2, 2, 3)
4. Let V = {ao+aj x+a2 x² | ao, a1, a2 E R} be a vector space of polynomial up to degree three, and p(x) = pPo+P1 x+p2 x², q(x) = qo+q1 x+q2 x² E V then show that = po.4o + pP1-q1 + P2-92 defines an inner product on 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.
: Discuss Application of inner product spaces with examples.
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

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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.