4. Let V be a vector space over C and let L: V → V be a linear transformation.State the dem(b) Let λ = C be arbitrary. State the definition of Vx, the X-eigenspace of L, and prove that itis a subspace of V.(c) Find the characteristic polynomial P₁(t) of the matrix A given below, and then the eigen-values of A.−1 -1 2A =0202 1-1(d) Choose one of the eigenvalues of A you found in (c), and find an eigenvector of A corre-sponding to this eigenvalue.

Answered: Image /qna-images/answer/e3c0f5be-ac57-42c6-b7a2-dacffdd0ce39.jpg

Answered: 4. Let V be a vector space over C and…

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 S P₂ P₂ be the linear transformation where S(x) = −12x − 30 and S(1) = 5x + 13. 0 Suppose Mg(S) = [-2] MB 0 3 Let Ɛ = {x, 1} be the standard basis of P2. Then P&B = 1 1 Then B = = {x+ x+ }
Consider a linear transformation T from R³ to R2 for which 0 3 6 T ([]) - · ~ ([1]) - · · (8) - C] = [{}], T = 8 2 T 0 = 5 Find the matrix A of T. A = =
Consider the linear transformation T: RR whose matrix A relative to the standard basis is given. 3 -1 A = 2 (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (21, 22) = 4,5 (b) Find a basis for each of the corresponding eigenspaces. B₁ = {(1,-1) B2 (1,-2) = (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). A' =
4. Let W be the xy-plane in R³. (a) Find an orthonormal basis ū₁, 2 for W. (b) Construct the matrix Q = [ ₁ ü₂ ] and calculate the matrix QQT. (c) Consider the linear transformation T: R³ → R³ that projects each vector orthogonally onto the xy-plane. Note that T() = QQT. Use the matrix to calculate T(₁), T(2), and T(ē3). (d) Explain how your answers make sense geometrically.
(1) For the following linear transformations T find its matrix representation with respect to the standard basis.
-9 V₁ = --[-] and X₂ such that T(x) is Ax for each x. Let x = A = X₁ -2 [3] and let T: R² R² be a linear transformation that maps x into x₁v₁ +X₂V₂. Find a matrix A -8 and V₂ =
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
Consider the linear transformation T: R → Rn whose matrix A relative to the standard basis is given. 2 A-[4] = 7 (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) -([ (2₁, 2₂) = (b) Find a basis for each of the corresponding eigenspaces. {[ B₁ B2 = A' = = (c) Find the matrix A' for T relative to the basis B', where B' is made up of the basis vectors found in part (b). 1
Let A be the matrix of the linear transformation T:R2→R2 that reflects each point across the x1-axis. What are the eigenvectors and corresponding eigenvalues of A?
Let L: R3 R2 be defined by the following. X1 + X3 ***] 8x2 Suppose X1 L X₂ P = = 1 -(HH) 4 3 3 1 2 B₁ = 4 ={[8][;}} is an ordered basis for the domain and B₂ = is an ordered basis for the range. Find the matrix representation P for L relative to B₁ and B₂ such that [L(u)]b₂ = P[u]ß₁'
1-5) First three drop box options (zero, negative, positive) Fourth box options (increase, decrease, constant) Last box (<,>)
defined by Let Let ƒ : R² → R² be the linear transformation [f] B = B C f(x) = = = 3 2 -2 - -3 x. be two different bases for R². Find the matrix [f] for f relative to the basis B in the domain and C in the codomain. {(-1,2), (3,-7)}, {{-1, -2), (3, 7)},

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