(8) Suppose that T'e C(V) for a finite-dimensional inner product space V (over C or R), and that ...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
3. Let V be a finite dimensional inner product space, and suppose that T is a linear operator on this space. (i) Let B be an ordered orthonormal basis for V and let U be the linear operator on V determined by [U19 = (T);. Then, for all 01,09 € V, (01, T(02)) = (U(V1), v2) (ii) Prove that the conclusion of the previous part does not hold, in general, if the basis 8 is not orthonormal.
Suppose V is a finite dimensional inner product space, and dim V = n. If is an orthogonal subset of V, prove that a. W can be extended to an orthogonal basis for V. b. is an orthogonal basis for c.
Please solve the math problem in detail. 8. Let V be a finite dimensional vector space over C, with a positive definite hermitian product. Let A: V→ V be a hermitian operator. Show that ltiA and 1-1A are invertible. [Hint: Ifu#0, show that IKHA) 8. Let V be a finite dimensional vector space over C, with a positive definite hermitian product. Let A: V→ V be a hermitian operator. Show that ltiA and 1-1A are invertible. [Hint: Ifu#0, show that...
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...
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)?...
Let V be a finite-dimensional inner product space, and let U and W be subspaces of V. Denote dim(V) = n, dim(U) = r, dim(W) = s. Recall that the proj and perp maps with respect to any subspace of V are linear transformations from V to V. Select all statements that are true. Note that not all definitions above may be used in the statements below If proju and perpu are both surjective, then n > 0 If perpw...
Let V be a finite-dimensional inner product space. For an operator TEL(V), define its norm by ||T|:= max{||Tull VEV. ||0|| = 1}. (1) To explain this, note that {l|Tu ve V, || 0 || = 1} is a non-empty subset of [0,00). The expression max{||TV|| | V EV, ||0|| = 1} means the maximum, or largest, value in this set. In words, the norm of an operator describes the maximal amount that it 'stretches' (or shrinks) vectors. (a) (1 point)...
3. Let V be a finite dimensional vector space with a positive definite scalar product. Let A: V-> V be a symmetric linear map. We say that A is positive definite if (Av, v) > 0 for all ve V and v 0. Prove: (a) if A is positive definite, then all eigenvalues are > 0. (b) If A is positive definite, then there exists a symmetric linear map B such that B2 = A and BA = AB. What...
just part c,d, and e please!! Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f eV" f(s) = 0 Vs ES}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and r & W, prove that there exists an few with f(x) +0. (c) If v inV, define u:V* → F by 0(f) = f(v). (This is linear...