課程名稱︰線性代數 課程性質︰工管系科管組必修 課程教師︰蘇柏青 開課學院：管理學院 開課系所︰工管系科管組 考試日期（年月日）︰2014.01.10 考試時限（分鐘）：應該也是三節課 試題 : 1. (45%) Let ╭ 2 1 2 ╮ │ 0 3 4 │ │ 0 4 0 │ C = [c1 c2 c3] = │ 0 0 0 │ ╰ 0 0 0 ╯ and let W = Col C = span {c1, c2, c3} 5 be the subspace of R for which the columns of C form a basis. (a) (10%) Perform Gram-Schmidt Process on {c1, c2, c3} and find an orthonormal basis {u1, u2, u3} for W (so that span {c1}= span {u1} and span {c1, c2}= span {u1, u2}). (If you get an orthogonal basis but not an orthonormal one, you'll get 7 points.) (b) (10%) Changing the order of the columns, perform Gram-Schmidt Process on {c1, c3, c2} and find an orthonormal basis {v1, v2, v3} for W (in this case, we want span {c1} = span {v1} and span {c1, c3} = span {v1, v2}). ⊥ (c) (10%) Find an orthonormal basis for W , the orthogonal complement of W. What is the dimension of W⊥? T (d) (5%) Let x= [1 2 3 4 5] . Try to decompose x into x = y + z so that y ∈ W and Z ∈ W⊥. (e) (10%) Find Pw, the 5 x 5 orthogonal projection matrix for W. T -1 T (Hint: Theorem 6.8 says Pw = C(C C) C, but this is not the only way you can find the answer. Note that W is not only span {c1, c2, c3}, but also span {u1, u2, u3} or span {v1, v2, v3}.) 2. (25%) Let M be the set of all real n x n matrices. In the class we n x n have shown that M is a vector space. n x n (a) (10%) Let S be the set of all symmetric matrices, i.e., n x n T S = {A∈M : A = A }. n x n n x n Show that S is a subspace of M . n x n n x n (Hint: A subset of a vector space is a subplace thereof if and only if it satisfies three conditions we discussed in class.) (b) (10%) Let L (M , M ) be the set of all linear transformation n x n m x m from M to M . Then (M , M ) is also a n x n m x m n x n m x m vector space. What is the dimension of L (M , M )? n x n m x m (No derivation is required.) (c) (5%) Let P3 be the set of all polynomials whose degrees are equal or 2 2 3 less than 3. We know that B = {1, 1+x, 1+x+x, 1+x+x+x } is also 3 a basis for P3. Express p(x) = x + x as a linear combination of elements in B. 2 -1 3. (30%) Let A = [ ]. 0 1 (a) (5%) Find the characteristic polynomial for A. (b) (2%) Find all eigenvalues of A. (c) (8%) For each of the eigenvalues of A, find the eigenspace corresponding to the eigenvalue. T (d) (15%) Repeat (a)-(c) for A. 考古題可以放三年，覺得自己也太會拖 好久沒打矩陣超麻煩，希望板主可以給些獎勵金 >~< --

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