課程名稱︰線性代數二 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06月19日(五) 考試時限：10:00-12:20，共140分鐘 試題 : 1. [15%] Let Y be a subspace of X. Recall that elements of the quotient X/Y are those subsets [x] = x + Y of X where x∈X. Let z1,...,zm∈X. Show the following. (a) {[z1],...,[zm]} generates X/Y if and only if Y + <z1,...,zm> = X. (b) [z1],...,[zm] are independent in X/Y if and only if z1,...,zm and Y are independent. (i.e., z1,...,zm are independent and <z1,...,zm>∩Y=0). (c) Let y1,...,yk∈Y. If {y1,...,yk} is a basis of Y and {[z1],...,[zm]} is a basis of X/Y, then {y1,...,yk,z1,...,zm} is a basis of X. 2. [15%] Let A∈Mn(R) be a symmetric matrix with eigenvalues λ1≧…≧λn. Let N∈M (R) satsifying N^t．N = Im and let μ1≧…≧μm be the n×m eigenvalues of the matrix N^tAN. Show that λi≧μi≧λ for all i=1,..,m. n-m+i 3. [20%] (a) Solve the differential equation d^2y dy ------ + 3 ---- + 2y = 0. dt^2 dt (b) Solve the linear system d (y1) ( 2 -1 -1)(y1) ---- (y2) = (-1 2 -1)(y2). dt (y3) (-1 -1 2)(y3) 4. [15%] Let S(t) = t^n +a_(n-1) t^(n-1) + … + a0∈R[t] be a monic polynomial. Suppose there is a factorization S = PQ for coprime polynomials P,Q. Show that if a function f(x) satisfies S(d/dx)f(x) = 0,then f = v+w for a unique pair (v,w) satisfying P(d/dx)v = 0 = Q(d/dx)w. 5. [10%] Show that there exist C∈Mn(C) and ε>0 such that for any self-adjoint matrix S∈Mn(C), the operator norm ∥C-S∥>ε. 6. [15%] Let V be a finite dimensional normed space and Ak,A∈L(V). Show that the sequence (Ak) converges to A for the operator norm if and only if for all x∈V, (Ak x) converges to Ax. 7. (a) [5%] Show that det(e^A) = e^(tr(A)) for any A∈Mn(C). (b) [10%] Let A∈Mn(C). Prove that all eigenvalues of e^A are of the form e^a, a an eigenvalue of A. (c) [10%] Is it true that every matrix B∈GLn(C) is of the form B = e^A for some A∈Mn(C)? Justify your answer. 8. [10%] Let V = R^n be the normed space using the norm (a1) |(︴)| = max |ai|. (an) i Let T = (t_ij)∈Mn(R) regarded as a linear operator T∈L(V). Show that the operator norm is n ∥T∥ = max Σ |t_ij|. i j=1 --

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