課程名稱︰工程數學─線性代數 課程性質︰電機系統一教學 課程教師︰ 開課學院： 開課系所︰電機系 考試日期（年月日）︰2009/6/17 考試時限（分鐘）：100 是否需發放獎勵金：來吧 （如未明確表示，則不予發放） 試題 : 1.Suppose the real n*n matrix A is invertible and diagonalizable, where n>1. Briefly explain why the following matrices are all diagonalizable: A^5+8A^2, T T ~ A^(-1), A , A A, and A obtained by 1) exchanging the first and last rows of A and 2) exchanging the first and last column of the resultant matrix. (25%) Four the these five matrices are always invertible and the other one may become not invertible. Indentify the one and explain why. (5%) n 2.In the real space R , define the inner product <x,y> = x1y1 + ... + xnyn T and the norm ||x|| = <x,x>^(1/2) for all vevtors x = [x1 ... xn] . n ┴ n Suppose S is a subspace of R . Let S be the orthogonal complement of S in R with respect to <‧,‧>. For any matrix A, let Null(A) denote its null space. Consider any real 4*4 matrix A with Null(A) spanned by the set T T T {[ 2 0 2 -1], [1 2 0 -1], [3 -1 4 -1] }. ┴ (a)Find an orthonormal basis for Null(A) .(10%) T (b)Find the vector z in Null(A) making ||z-[1 1 1 1] || the smallest. (10%) (hint: It is easier to use the result of (a).) T (c)Find the set of all real 4*4 matrix A = A having the above Null(A). (10%) (hint: recall the Spectral Decomposition Theorem.) 4 (d)With the above Null(A), let any v ∈ R be uniquely decomposed as v = u + w, ┴ where u ∈ Null(A) and w ∈ Null(A) , and define a linear operator T: 4 4 R → R as T(v) = u - w. Show T is orthogonal. (10%) Also determine all eigenvalues of T. (5%) (hint: consider the proper equivalent condition for the orthogonality of T.) T 3.Let V be the real vector space with vectors [ p1(x) p2(x) ] , where p1(x) and p2(x) are polynomials in x with degrees no larger than 2. For vectors T T T T [ p1(x) p2(x) ] and [ q1(x) q2(x) ] in V, [ p1(x) p2(x) ] + [ q1(x) q2(x) ] = T [ p1(x) + q1(x) p2(x) + q2(x) ] . T T For r∈R, r[ p1(x) p2(x) ] = [ rp1(x) rp2(x)]. (a)Find a basis B of V. (10%) (b)Consider the linear operator D: V → V defined by T T D( [ p1(x) p2(x) ] ) = [ dp1(x)/dx dp2(x)/dx ] . Find [D]B and decide if D is one-to one or onto. (15%) --

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