For each of the following linear systems, discuss the nature of the origin and sketch a correspondingphase portrait (including an explicit computation of the eigenvectors in the case of real eigenvalues).If the origin is a saddle point, identify its stable and unstable manifold.1.2.3.4.5.[x = 4x - y= 2x + yx=5x-3yy = 3x - 5yx = = 5y - xy = - Yx = -4x - 5yy = 4x + 4yx = 3x - 4yy = 2x - y

The given system is x.=4x-yy.=2x+y. Sketch a phase portrait. Also, we have to find the eigenvalue…

Answered: For each of the following linear…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.1: Eigenvalues And Eigenvectors

Problem 77E

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Please do e and f for 6.2.1 And write it out
Suppose T: R² → R2 is the linear transformation defined in the figure below. The figure shows where T maps 8 vectors V₁,..., vg from the domain. With this limited information about T, what properties of T can be determined? У У -1 -2 -3 -4 v6 8 7 6 5 v4 4 3 2 1 v5 * v1 v3 v2 8 7 6 5 4 3 21 T(v2) T(v3) T(v4) T * T(v1) -1 -2 T(v8) T(v7) T(v6) -3 T(v5) x v7 v8 -5 -5 -6 -6 -7 -7 -8 -8 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 domain codomain
Q1. a. Find the eigen values of the following matrix and discuss the applications of eigen values in engineering disciplines. 2 5 4 A=0 6-3 00 8 b. Temperature of a disk brake plate at any point (x, y) varies is represented by the T(x,y)=100/(1+x³+y³ ) where T measure in °C and x, y in meters. Find the rate of change of temperature with respect to x direction and y direction and also the rate at a point (2,1).
q14. true or false ?
-2 2. Find h in the matrix A = 7 such that the eigenspace for 1 = 7 is one- -1 4 -6 5 7 dimensional. Find h such that the eigenspace for 1=7 is two-dimensional.
a.) The function, f(x) = 3X2-1, is an eigenfunction of the operator, (see image attched). Find the eigenvalue corresponding to this eigenfunction. b.) Show that the function is an eigenfunction of the operator, (See image attached) and calculate the eigenvalue.
in pic
Do b and d for 6.2.1 Please write out
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need help figure out As two equations: (write "c1" and "c2" for c1c1 and c2c2 for x(t) and y(t)
Use (a) to solve the system x = (: :) . Х. -8 7 NB: Real solutions are required. Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly independent solutions. The second will yield solutions which are identical (up to a constant) to the solutions already found. a) is as follows Show from first principles, i.e., by using the definition of linear independence, that if x+iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector v = u + iw, then the two real solutions Y(t) = et (u cos bt – w sin bt) and Z(t) = eat (u sin bt + w cos bt) are linearly independent solutions of X = AX.
Suppose that the matrix A has the following eigenvalues and eigenvectors:

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