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.

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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 Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D : P3 - P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) = P3. Is D a linear transformation? Let p(x) = a³x³ + ²x² + a₁ + a₁ and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = = Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ✪ c. Is D a linear transformation? choose ◆
Diagonalize the quadratic form by finding an orthogonal matrix Q such that the change of variable x- Qy transforms the given form into one with no cross-product Enter your answer in the form (Q. f(y)) = ([[row 1], [row 2]]. (y)). where each row is a comma- C terms. Give Q and the new quadratic form, f(y). (Assume y separated list and f(y) is in terms of y (Q. (y)) - 6x₁² + 9x₂² Need Help? - 4x₁x2 Read it and Y₂)
4. Let V be a vector space and let L: V → V be a linear transformation. Let A be a scalar. (a) Define VA, the A-eigenspace, and prove that it is a subspace of V. (b) Let v and v2 be eigenvectors with eigenval- ues A1 and A2, respectively. If A1 # 12, prove that v, and V2 are linearly independent.
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.
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?
4. Let W be the subspace of R³ defined by the equation 2x – y + 3z = 0. (d) Let T: R³ → R³ be the linear transformation given by reflection across W. Determine the B-matrix of T. Use the result of part (d) to find the matrix of T with respect to the standard basis.
Consider the following matrices: √/3 Q 8 M' = – 051010 100 10+ 9 (a) Find the component q in the coordinate transformation matrix Q such that the matrix is orthogonal and associated with rotation. q= = (b) Let Q transform the orthonormal basis set B to B'. Determine the components of matrix M' in the new orthonormal basis set B'. (c) Calculate the three invariants of M. I₁ I₂ = M = (d) Calculate the three invariants of M'. I I! = [2 −1 6 7 -5 6 1 -1 4 I3 = =

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