6. Define f R2nR via the inner product as follows. Writing points of R2n as (x,...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
(9) Let E R" and let A E L(R"). Define a map f : R" -> R" by f (x) A,)v. Here (is the Euclidean inner product (a) Prove that f is a C1 map and find f'(x) (b) Prove that there exist two that f U V is a bijection on R" neighborhoods of the origin in R", U and V, such
(9) Let E R" and let A E L(R"). Define a map f : R" -> R"...
3) Define the relation <on R via x < y if and only if xy < 10. Show that is symmetric. (20 points)
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...
Upts) GIve the text of the Spectral Theorem on a real inner product space E (3pts) Prove that any eigenvalue of a self-adjoint linear map on a complex inner product space is real. 4,) (3pts) Give the definition of a skew-symmetric matrix. X Lexercisebethe car points baseofPandaERaparameter -C )ER . For all = ( 1 ) E R3 and y-(h /2 yE R2 we define the bilinear form ba by 4 y. (3pts) For which value of a, b, is...
3. Give P an inner product structure by defining (f,9)-of(ag(x)e"da (you do not have to show that this is an inner product). Apply the Gram-Schmidt process to (1, , ) with this inner product. Show your work. Hint: You may want to use the fact that Jooz"e-xdz = nl, which follows from n application of integration by parts. (You do not have to show this.) Fun Fact: The polynomials you find here are the first three Laguerre Polynomials, and are...
Definition 0.1. The inner product is a map <-, - >: V x V +R where V is a vector space satisfying (1) Conjugate symmetry < x, y >= <Y, X > For us in the reals, ignore the complex conjugate. (2) Linearity in the first argument <ax, y >= a< x, y> and < x + y, z >=< 3,2 >+<y, z> (3) Positive definiteness < x, x > 0 and < x, x >= 0 & x=0 The...
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Question 3. Let 3 5/' and for x(2),y -(,) ER2 define (a) Show that the assignment (x, y) > (x,y) defined ın (1) us an nner product [10 marks (b) If a - (1,-1) and b - (1,1), then show that the vectors a and b are lınearly ndependent but they are not orthogonal with respect to the inner product n (1) 3 marks] (c) Given the vectors a and b in (b), the set (a, by is hence a...
2. Let ro < 1<..< n be n + 1 distinct points in IR. Define polynomials Co, ..., (n of degree n by (r - k) Let P, = 1,[r] be the polynomials of degree n, which is a vector space of dimension n + 1. (a) Show that the n+1 polynomials {lo, ..., Ln^ are basis for P i.e., they are linearly independent. (b) Find the coordinates [f]в of polynomial f E 1, with respect to the basis l-[10,...