Suppose TE L(V) and By : v1, V2, ..., Vn is a basis for V such that for By3 2 1]0 1 00 0 2M(T)(a) What is dim(V), ie what is n?(b) Without doing any calculations, what are the eigenvalues of T? Explain howyou know.(c) Suppose A is an eigenvalue for the matrix M(T) and u e F31 is aneigenvector associated with A, ie forabe F31.aM(T)u = Au = A bProve that av1 + bv2 + cv3 is an eigenvector for T corresponding toeigenvalue A. (Hint: Recall Result 3.65 in the textbook which can also bewritten as: [Tu]By = M(T)[v]Bv for T E L(V). Also note that the linearmap that takes v e V to [v]By € Fl is an isomorphism.)(d) Use the conclusion of part (c) to find the eigenspaces of T.(e) Is T diagonalizable?

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Answered: Suppose TE L(V) and By : v1, V2, ...,…

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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3. If A = 0 -1 0then the bases for the eigenspaces of A2 + 21 are: 2. a) ({, = 7, v, = (1,0,0), {A2 = 4, v (-1,0,1)}, and (A = 1, v, = (-1,6,0}) b) {{ =-3, v, = (-1,6,0}), (A = 3, v (1,0,0}}, and (A = 0, v, = (1,0,0}] c) [{ = 27, v, = (1,0,0},{22 = 6, v = (-1,0,1}}, and (As = 3, s =(-1,6,0}} d) {{2 = 2, v, ={-1,0,1}}. [A2 = 23, , = [1,0,0), and (As =-1,, {-1,6,0}}
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5. Let A be a 6 x 6 matrix with characteristic equation X2(a – 1) (A – 2)3= 0. What are the possible dimensions for eigenspaces of A?
5. Let T: R2 → R2 be defined by T = and matrix A -x y ( []) = [- ² + 1] - [TB where B is the standard basis for R2 = i) Find the eigenvalues of A ii) Find the corresponding eigenvectors F [2 2 [² √3 = [21] = V. B
Chads baseball team is practicing hitting in the batting cages for pitching machines pitch baseballs at the rate of 35 pitches in 5 minutes chad practiced hitting for 8 minutes how many baseballs were pitched to chad in that time
6. Let A be an n x n hermitian matrix, i.e. A = A*. Assume that all n eigenvalues are different. Then the normalized eigenvectors { v; : j = 1,...,n} form an orthonormal basis in C". Consider B:= (Ax – ux, Ax – vx) = (Ax – ux)*(Ax – vx) | where (, ) denotes the scalar product in C" and µ,v are real constants with µ 0.
of Let W be the set of all 2 × 2 matrices [a b] C d basis for W is S = (a) • X (b) following questions: • Y = • r {[ || (c) where, c = 0 and d = 2a - 3b. A || •S= 1 x ] []} " S О У I dim(W) - Answer the
In each case, determine whether the vector v is an eigenvector of the 3 by 3 matrix A. No A = No A = No A = 9 -5 18 -8 6 -18 -4 5 -13 -7 -2 3 17 2 -13 -12 -2 8 -1 -4 -8 10 3 -2 -5 -4 -4 1. V = 2. " 3. υ: || 3 7 1 -1 1 -1 7 B 4 1

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