課程名稱︰線性代數一 課程性質︰數學系必修 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︰2019年11月08日，10:00-12:10 考試時限：130分鐘 試題： In the following, F denotes a field. 1. [20%] Let V be the vector space over |R consisting of all polynomials c0 + c1 x +c2 x^2 of degree ≦ 2 with real coefficients. Let f1, f2, f3 ∈ V* defined for p ∈ V by 1 2 -1 f1(p) = ∫p(x)dx,　　f2(p) = ∫p(x)dx,　　f3(p) = ∫p(x)dx. 0 0 0 (a) Find a basis {p1,p2,p3} of V such that {f1,f2,f3} is the dual basis. (b) Let D : V → V be the linear map defined by taking derivative D(p)=p'. Let D^t : V* → V* be the transpose of D. Describe D^t(f1) explicitly. 2. [15%] Let A, B ∈ M_n(|R). Prove the following. (a) If trace(AA^t) = 0, then A = 0. (b) trace(AB) = trace(BA). (c) AB - BA = I is impossible. 3. [10%] Let W1 and W2 be subspaces of a vector space V such that the union W1 ∪ W2 is also a subspace. Show that either W1 ⊂ W2 or W2 ⊂ W1. 4. [15%] Let W1 and W2 be subspaces of a vector space V. Show that the following are equivalent. (a) W1 + W2 = V and W1 ∩ W2 = {0}. (b) For any v ∈ V there is a unique pair of vector w1 ∈ W1, w2 ∈ W2 such that v = w1 + w2. 5. [10%] Suppose A ∈ M_n(F) is not invertible. Show that there exists B ∈ M_n(F) satisfying B ≠ 0 and AB = 0. 6. [10%] Let T : V → V be a linear operator. Prove the following are equivalent. (a) The intersection of the image of T and the kernel of T equals {0}. (b) If T(Tv) = 0, then Tv = 0. 7. [10%] Let V be an n-dimensional vector space and T : V → V be linear. Suppose T^(n-1) ≠ 0 and T^n = 0. Show that there exists v ∈ V such that v, Tv, T^2v, ..., T^(n-1)v form a basis. 8. [10%] Let A ∈ M_(m×n)(F). Show that the row reduced echelon form of A is unique: if P, P' ∈ M_m(F) are invertible such that PA = P'A are row reduced echelon form of A, then PA = P'A. 9. [15%] Let V be a finite dimensional vector space and T ∈ L(V). Suppose T^2 = T. Let 1_V ∈ L(V) be the identity transformation and let S = 1_V - T. (a) Show that S^2 = S. (b) Show that ker(S) = im(T) and im(S) = ker(T). (c) Show that V = ker(T) + ker(S) and ker(T)∩ker(S) = {0}. 10. [10%] Let W = M_n(F) and W0 the subspace spanned by matrices C of the form C = AB - BA. Prove that W0 is exactly the subspace of matrices which have trace zero. 10. --

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