課程名稱︰線性代數一 課程性質︰數學系大一必修 課程教師︰余正道 開課學院：理學院 開課系所︰數學系 考試日期︰2020年01月03日(五)，10:00-12:10 考試時限：130分鐘 試題 : There are 9 problems in 2 pages. In the following, F denotes a field. 1. [10%] Let A∈M_n(F). If r is any positive integer with 1≦r≦n, an r × r submatrix of A is any r × r matrix obtained by deleting (n-r) rows and (n-r) columns of A. Suppose A≠0. Prove that the rank of A equals the largest positive integer r such that some r × r submatrix of A i invertible. 2. [10%] Let A∈M_n(F). Suppose A is diagonalizable and has characteristic polynomial d_1 d_k (x-c_1) … (x-c_k),　　　c_i ≠ c_j. Let V be the space of B∈M_n(F) such that AB = BA. Prove that dim(V) = d_1^2 + ... + d_k^2. 3. [20%] Let T∈L(V) where V is finite-dimensional. (a) Suppose p(x), q(x)∈F[x] are coprime (i.e., gcd(p,q) = 1) with the characteristic polynomial ch(x) = p(x)q(x). Let W be a T-invariant subspace. Show that W = (W∩Ker p(T)) ⊕ (W∩Ker q(T)). (b) Let W be a T-invariant subspace. Suppose T is diagonalizable. Show that the restriction operator T_W∈L(W) is diagonalizable. 4. Let V be an n-dimensional vector space and T∈L(V). Suppose T is diagonalizable. (a) [5%] If T has a cyclic vector, show that T has n distinct eigenvalues. (b) [10%] If T has n distinct eigenvalues and if {v_1,...,v_n} is a basis of eigenvectors of T, show that v = v_1 + ... + v_n is a cyclic vector for T. ( 1 2 1 0 ) 5. [10%] Let A = ( -2 1 0 1 )∈M_4(R) whose characteristic polynomial is ( 0 0 1 2 ) ( 0 0 -2 1 ) x^4 - 4x^3 + 14x^2 - 20x +25 = (x^2 - 2x + 5)^2. Find an invertible matrix P and the rational canonical form Q of A such that P^(-1) AP = Q. 6. [10%] Let A∈M_n(R) satisfying A^2 + I_n = 0. Prove that n is even, and if n = 2k, then A is similar to ( 0 -I_k ) ( I_k 0 ). 7. (a)[10%] Prove that every complex n × n matrix is similar to its transpose. (b) [5%] Ture or false: for any field F, every A∈M_n(F) is similar to A^t. Justify your answer. 8. Let A, B∈M_n(F). (a) [5%] Prove that if (I-AB) is invertible, then (I-BA) is invertible with (I-BA)^(-1) = I + B(I-AB)^(-1) A. (b) [10%] Prove that AB and BA have precisely the same eigenvalues. (c) [10%] Do AB and BA have the same characteristic polynomial? Do they have the same minimal polynomial? 9. [15%] Let V be an n-dimensional vector space and T∈L(V) with T^(k+1) = 0 for some integer k≧0. Let S_r = T^r for r≧0. Show that there exists uniquely a chain of subspaces V = W_k ⊃ W_(k-1) ⊃ ... satisfying (i) T(W_i) ⊂ W_(i-2), (ii) S_r(W_r) = W_(-r), and -1 (iii) W_r ∩ S (W ) = W . r -r-1 r-1 --

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