課程名稱︰線性代數 課程性質︰系必修 課程教師︰蘇柏青 開課學院：電資學院 開課系所︰電機系 考試日期（年月日）︰2015/5/21 考試時限（分鐘）：50 min 試題 : Linear Algebra Quiz #2 Thursday May 21, 2015 1. Eigenvalues, Eigenvectors, and Diagonalization (30%) _ _ | 3 -1 | Let A = | | | -1 3 | - - (a) (4%) Find the characteristic polynomial of A. (b) (4%) Find eigenvalues of A. Indicate the multiplicity for each eigenvalue. (c) (4%) For each of the eigenvalues of A, find a corresponding eigenvector. (d) (4%) Find the eigenspace corresponding to each eigenvalue. Also indicate the dimension of each of these eigenspaces. (e) (6%) Is A diagonalizable? If so,find an invertible matrix P and a diagonal matrix D that satisfy D = P^-1 A P. (f) (4%) Is A orthoganally diagonalizable? That is, does there exist an orthogonal matrix Q so that Q^T A Q is diagonal? (g) (4%) Calculate A^6. 2. Gram - Schmidt Process (20%) For each the following bases for R^3 , determine whether it is orthogonal, orthonomal, or neither. If it is not orthonormal, apply the Gram-Schmidt process and vector-length normalization to obtain an orthonormal basis. (a) (5%) S1 = {[ 1 0 0 ]^T , [ 0 1 0 ]^T , [ 0 0 1 ]^T}. (b) (5%) S2 = {[ 1 0 0 ]^T , [ 0 2 0 ]^T , [ 0 0 3 ]^T}. (c) (5%) S3 = {[ (sqrt2)/2 (sqrt2)/2 0]^T,[(sqrt2)/2 -(sqrt2)/2 0^T],[0 0 3]^T} (d) (5%) S4 = {[ 1 1 0 ]^T , [ 1 1 -1]^T , [ 1 1 -1]^T}. 3. Matrix representation of Linear Operators (50%) Let T : R^3 -> R^3 be defined as _ _ _ _ | x1 | | x1 + x3 | T( | x2 | ) = | x2 - 2x3 | | x3 | | 6x1 - x2 + 3x3 | - - - - _ _ _ _ _ _ | 0 | | 1 | | 1 | and let B = {| -1 | , | 0 | , | -1 | } be a basis for R^3. | 1 | | -1 | | 1 | - - - - - - (a) (2%) Find A , the standard matrix of T. _ _ | 1 | (b) (9%) Let v = | 0 | . Calculate [v] , T(v) and [T(v)] . | -1 | B B - - (c) (9%) Find [T] , the matrix representation of T with respect to B. B (d) (5%) Determine whether 5 is an eigenvalue of A by claculating det( A - 5I). If so , find the eigenspace of A corresponding to the eigenvalue 5. (e) (5%) Find the characteristic polynomial for A. (f) (5%) Is A diagonalizable? If so, find an invertible matrix P and a diagonal matrix D1 such that A = P D1 P^-1. (g) (5%) Determine whether 1 is an eigenvalue of [T] by calculating det([T]-5I) B B If so, find the eigenspace of [T] corresponding to the eigenvalue 5. B (h) (5%) Find the charateristic polynomial for [T] . B (i) (5%) Is [T] diagonalizable?If so,find an invertible matrix Q and a diagonal B matrix D2 such that [T] = Q D2 Q^-1. B --

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