Problem 5Short answers: (No explanations needed. Simply write your answers. If you dosome calculation to get the answer, show the calculation.)(a) If 4 is an eigenvalue of a matrix A, then what is det (A-41)?(b) Suppose T: R¹ R³ is an onto linear transformation. What is the dimension of rangeof T?

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Answered: Problem 5 Short answers: (No…

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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Page 1: 1 as a single matrix.) 5. A linear transformation T has eigenvalues 2 and 5, where 2 has multiplicity 2. Solving the matrix equation yields the solution set X yields the solution set Y Solving the matrix equation (T-21) = 0 = x 4 (T – 51)v = Ő y 6s+ 1 t. 5 = 0 t. 2 If T can be diagonalized, write the diagonalization (you do not need to simplify any inverses). If T cannot be diagonalized, explain why.
1 1 3 Consider the matrix A = -2 8 4 -1 Determine whether the matrix is invertible or not using only observable properties of the matrix, that is, no calculation. Explain how your arrived at your determination. In light of your previous answer, what can be said about the linear transformation 7H Aã? Why? What can be said about Col A and Nul A in this case? Why? Sketch a function picture indicating the domain and codomain. Then identify Col A, identify Nul A in your picture and indicate where Nul A maps to.
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. - 2 - 5 -6 11 -3 ; λ 2,4,5 - 6 2 ..... Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. 2 0 0 For P = D= 0 4 0 0 0 5 (Simplify your answer.) B. The matrix cannot be diagonalized.
For the system of equations ()- dæ -3 -1 x. dt 1 1. Find the eigenvalues and eigenvectors of the matrix. 2. Find the coordinate transformation that will put the equation into canonical form and show that this works. 3. Write down the general solution.
Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 10 and -9. A = 1 -3 9 -3 93 9 3 1 Enter the matrices P and D below. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)
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4. Let E: P3 R2 be given by E (p(x)) = [P(2)] (a) Find a matrix that induces the transformation E. TH (b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible. (c) Find a polynomial q(2) where E(q(z)) = [], or explain why this isn't possible. 8P (d) Find a nonzero polynomial r(z) where E(r(2)) = [] or explain why this isn't possible. (e) Describe the kernel of E. What is the dimension of the kernel? Find a basis for the kernel. (f) Describe the image of E. What is the dimension of the image? Find a basis for the image. (g) Is the transformation E one to one? Is it onto? Is it an isomorphism? Explain.
If the matrix that represents a rotation of π/3 about the origin is: ( a b c d) What are the values of a, b, c and d? Give your answers exactly, not as decimal approximations.
Find the standard matrix of each of the following linear transformations. (a) T: R² R², which reflects each point in R² across the line x₁ = 0 (i.e. the 22-axis) - -> (b) S: R² R², which rotates each point in R² clockwise by 90° and then dilates it by a factor of 7 (c) P: R³ R³, which projects each point in R³ onto the plane z₁ = 0 (i.c. the 2x3-plane)
Circle correct answers: If for a given matrix M you built a matrix of unit eigenvectors (matrix C), then the (normality, congeniality, similarity, differentiality) transformation C1 M C results in (normalization, diagonalization, orthogonalization, inversion, deformation) of the matrix M.

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