MATH_215_Test3

# MATH_215_Test3 - part(a Find the particular solution that...

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Practice Test 3 For full credit, show all work. You can use your calculator for com- putational purposes. But your work on paper must be transparent enough that I understand your answer without a calculator. No credit will be rewarded if I cannot understand how you obtain a numerical answer. 1 Deﬁne: Note: these are just samples. a) Eigenvalues and eigenvectors for a matrix A . b) Similarity transformation. c) General vector space V . 2 Consider the matrix A = 2 1 - 4 0 0 - 1 0 1 - 2 . a) Using elementary row operations and row echelon form of A , compute the determinant of A . b) Find the eigenvalues for A and compute the respective eigenspace. De- termine the algebraic and geometric multiplicities for each eigenvalue. 3a) Consider the matrix A = - 1 0 2 2 1 - 2 0 0 1 . Find the similarity trans- formation P that diagonalizes A . Write down the diagonalization of A . b) Consider the system of linear equation x 0 = A x with A as deﬁned in

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Unformatted text preview: part (a). Find the particular solution that satisﬁes the initial condition x = 1 3 . 4 Consider the equation for linear dynamical system x k +1 = A x k where A = ± 1 2-2 1 ! . Factorize A into ± r r !± cos θ-sin θ sin θ cos θ ! by com-puting r and θ . Characterize the zero ﬁxed point. Explain. 5a) Show that B = { 1-x, 1 + x 2 , x-x 2 } forms a basis for P 2 ( R ). b) Let C = { 1 , 1 + x, 1 + x + x 2 } be another basis for P 2 ( R ). Find the change of basis matrix from B to C , using the standard basis as an intermediate basis. 6 Prove the following statements: a) Prove that det( kA-λI ) = k n det ( A-λ k I ) . b) Let A and B be n × n matrices, each with n distinct eigenvalues. Prove that A and B have the same eigenvectors if and only if AB = BA ....
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MATH_215_Test3 - part(a Find the particular solution that...

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