Zet <- be the usual inner product on₂.)the complex vector space (₂[x]. Recallthat C is a 1-dimensional vector space over C.Consider the function To C₂ [x] → C given by T(g(x)) =A) Prove that T is a linear transformation.B) Find a basis for the kernel of T.

Given : T : ℂ2x → ℂ defined by Tgx = g(x), x, where . , . is the usual inner product on ℂ2x. To…

Answered: Let (..) be the usual inner product on…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let V be a vector space, v, u E V, and let T₁ : V → T₁ (v) = 5v +6u, T₂ (v) = 4v 4u, Find the images of v and è under the composite of T₁ and T₂. (T₂T₁)(v) = (T₂T₁)(u) = - V and T₂ : V → V be linear transformations such that T₁(u) = −3v - 3u, T₂(u) = 6v+6u.
Consider the vector space F of all functions from R to R with pointwise addition and scalar multiplication. (a) Show that the three vectors fi = e“, f2 = xe", f3 = x²e" are linearly independent in F. (b) Consider the subspace H = span{f1, f2, f3}. Let T : H → H be the linear transformation defined by df T(f)= + f. dx Find the matrix of T' with respect to the basis B = {fi, f2, f3} of H (i.e., find 8[T]B). (c) Is there a basis C of H in which c |T|c is diagonal? Explain why or why not.
Define T: R² R² by T() = T 21 =[ =[ U2 that T is a linear transformation. ● a) Let u = I1 and 7 = X2 = 3x12x2 2x2 V1 be two vectors in R2 and let c be any scalar. Prove U₂
Let P2(R) be the vector space of polynomials over R up to degree 2. ConsiderB ={1 + x − 2x^2 , −1 + x + x^2, 1 − x + x^2},B' ={1 − 3x + x^2, 1 − 3x − 2x^2, 1 − 2x + 3x^2}.(a) Show that B and B' are bases of P2(R).(b) Find the coordinate matrices of p(x) = 9x^2 + 4x − 2 relative to the bases B and B'.(c) Find the transition matrix P_B=→B'(d) Verify that, [x(p)]B'= P_B→B' [x(p)B].
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.
-(.. Suppose that T: V V is a linear transformation and V is a finite dimensional vector space. Suppose that U is a subspace of V such that T(U) = T(V). Prove that U + ker(T) = V. %3D

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Let (..) be the usual inner product on the complex vector space (₂[x]. Recall that is a 1-dimensional vector Space over C. Consider the function To C₂[x] → C given by T(g(x)) = (g(x), x) A) Prove that T is a linear transformation. B) Find a basis for the kernel of T.