課程名稱︰線性代數 課程性質︰數學系必修 課程教師︰林紹雄 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2011/11/5 考試時限（分鐘）：180分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : There are problems A to F with a total of 140 points. Please write down your computational or prrof steps clearly on the answer sheets. A. Consider the matrix ┌ 0 -1 1 3 3 ┐ │-1 1 1 2 3 │ A = │ 0 2 -2 -6 -6 │ └ 0 0 1 1 2 ┘ (a)(10 points) Find its LDU-decomposition in the form PA=LDU, where P is a permutation matrix. Find the rank and nullity of A. (b)(8 points) From PA=LDU, write down a basis for the four fundamental subspaces of A. 4 (c)(7 points) Given b∈R , find conditions on b so that Ax = b is solvable, and find the general solutions. ------------------------------------------------------------------------- B.(15 points) Consider the square matrix (where a, b, c, d are real) ┌ a a a a ┐ │ a b b b │ A = │ a b c c │ └ a b c d ┘ -1 Determine the conditions on a,b,c,d s.t. A is invertible, and find A . When A is not invertible, find its rank and nullity. Do you need permutation matrix to perform LDU-decomposition of A? ------------------------------------------------------------------------- x C. Consider the real vector space P = { p(x) e | p(x) is a polynomial 3 of degree <= 3}, and T:P →P defined by T(f) = f' - f. 3 3 (a)(6 points) Find ker(T), and range(T), and verify the nullity theorem. (b)(14 points) Find the matrix A which represents T with respect to the basis 3 x 3 x 3 x x f1(x) = (x-1) e , f2(x) = (x+1) e , f3(x) = x e , f4(x) = e and the matrix B which also represents T with respect to the basis 3 x 2 x x g1(x) = (x +x+1)e , g2(x) = (x +x+1)e , g3(x) = (x+1)e , x g4(x) = e . Are A and B similar matrices? Prove your answer. ------------------------------------------------------------------------- D. Given the matrix ┌ -1 1 0 0 ┐ │ -1 0 1 0 │ A = │ 0 -1 1 0 │ │ 0 -1 0 1 │ │ -1 0 0 1 │ └ 0 0 -1 1 ┘ (a)(10 points) Draw the simple connected graph G whose incidence matrix 6 is A. Apply KCL to write independent conditions on b∈R s.t. b lie in the column space of A. How many independent loops are there in G? (b)(10 points) The path T (in the underlying graph of G) given by vertex 1 → vertex 4 → vertex 3 → vertex 2 is a spanning tree of G. Find a LDU-decomposition of A in the form PA=LDU so that the first 3 rows of U is the incidence matrix of T. ------------------------------------------------------------------------- E. Prove the following statements. Each has 15 points. (a) Given a block matrix of the form Q = ┌ Ik B ┐ Prove that Q is └ C Il ┘ invertible iff Ik - BC is invertible iff Il - CB is invertible. -1 Express Q in terms of B, C explicitly. (Ik, Il就是k*k, l*l的單位矩陣) (b) Let C be a binary linear code, and define ε = { c∈C | c has even weight}. Prove that ε is also a linear code, and when ε≠ C, the number of elements of ε is exactly half of the number of elements in C. ------------------------------------------------------------------------- F. Determine which of the following statement is true. Prove your answer. Each has 6 points. (a) Let V be a vector space over R. S, T: V → V are two linear transformations. If T(S(v)) = 100v for all v∈V, then T is onto, and hence T is invertible. (b) Any [10,7,d]-binary linear code is not 1-error correcting, but is possible to be 1-error detecting. T T (c) A matrix A has full rank iff either AA is invertible, or A A is. (d) If a matrix as a LU-decomposition, then its LU-decomposition must be unique. 8 (e) There exist three 6-dimensional subspaces U, V, W of F such that dim(U∩V∩W) = 1. -- --

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