課程名稱︰工程數學-線性代數 課程性質︰電機系必修 課程教師︰劉志文 開課學院：電資學院 開課系所︰電機系 考試日期（年月日）︰109.01.10 考試時限（分鐘）：100 試題 : 1.(10%) Probe that any orthogonal subset of R^n consisting of nonzero vector is linearly independent. 2.(25%) The "Orthogonal Decomposition Theorem" is stated as follows. Let W be a subspace of R^n. Then, for any vector n in R^n, there exists unique vector w in W and z in W perp such that u = w + z. In addition, if {v1, v2, .. ., vk} is an orthonormal basis for W, then w = (u‧v1)v1 + (u‧v2)v2 + ... + (u‧vk)vk. (A)(5%) Show that w=Pw u, where Pw is the orthogonal projection matrix on W and Pw = v1v1^perp + v2v2^perp + ... + vkvk^perp. (B)(5%) Define vivi^perp = Pi, prove that rank(Pi) = 1, PiPi = Pi, for i = 1 ..., k, and PiPj = 0 (zero matrix) for i is not equal to j. (C)(5%) Prove that Pw = P1 + P2 + ... + Pk is a symmetric matrix and Pw^2 = Pw (D)(5%) Find the eigenvalues of Pw and their corresponding eigenspaces. (E)(5%) Fint the orthogonal projection matrix on W perp, P_{w perp}, in temrs of Pw and In which is nxn identity matrix. 3.(15%) Let T be the linear operator on R^n defined by ┌x1┐ ┌-x1 ┐ │x2│ │ x2 │ T(│x3│) = │ x3 │ │. │ │ . │ │. │ │ . │ │. │ │ . │ └xn┘ └ xn ┘ (A)(5%) Show that T is an orthogonal operator. (B)(5%) Show that the only eigenvalues of T are 1 and -1, respectively. (C)(5%) Let W be the eigenspaces of T corresponding to eigenvalue 1. Let Q be the standard matrix of T, and Pw be the orthogonal projection matrix on W. Show that Q = 2Pw - In, where In is nxn identity matrix. 4.(12%) Let ┌-5 6 1┐ A = │-1 2 1│ └-8 6 4┘ where eigenvalues are 2, 3, and -4. (A)(6%) Please find the eigenvalues of A^-1. (B)(6%) Please find the eigenvalues of A^2 + 2A. 5.(8%) A square matrix A is a nilpotent matrix if there exists a positive int- eger p such that A^p = O. Show that the only eigenvalue of A is 0. 6.(10%) Determine all values of a such that the following matrix is NOT diago- nalizable. ┌-1 0 0┐ │ 3 -2 1│ └ 0 0 a┘ 7.(5%) In the vector space of real-valued function F = {f│f:R→R}, determine if the following set S is linearly independent. S = {sin^2x, cos^2x, 2} 8.(10%) Let M_2x2 be the set of all 2x2 matrices. Let T be the function on M defined by T(A) = (traceA)┌1 2┐. If A = ┌a11 a12┐, trace A = a11 + a12. └3 4┘ └a21 a22┘ (A)(5%) Prove that T is linear. (B)(5%) Supposed that B is the basis for M, B = {┌1 0┐,┌0 1┐,┌0 0┐,┌0 0┐}. └0 0┘ └0 0┘ └1 0┘ └0 1┘ Determine [T]_B. 9.(5%) Let L(V,W) be the set of all linear transformations from vector space V to vector space W. Suppose V is the set of all mxn matrices, V = M_mxn and W is the set of all linear transformations from R^n to R^m, W = L(R^n,R^m). Please find the dimension of L(V,W). --

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