课程名称︰线性代数二 课程性质︰数学系大一必修 课程教师︰庄武谚老师 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2019/6/5 (三) 考试时限（分钟）：150 试题 : (There are totally 115 points) (以下ε代表属於符号) (1) (15 points) Let V be a finite dimensional vector space and W be a subspace of V. Show that(V/W)* is isomorphic to {lεV*| l restrict to W=0} as vector spaces (2) (15 points) Let AεMnxn(C) be a square matrix with a polar decomposition A=WP, where W is unitary and P is positive semidefinite. Show that A is normal if and only if WP^2=P^2W (3) (15 points) Find a singular value decomposition for the following matrix 2 -1 0 0 (1 1 1 1) 1 1 -1 -1 (4) (15 points) Let (V,H) be a quadratic space. Suppose that x,yεV are anisotropic vectors satisfying H(x,x) = H(y,y). Show that there exists an isometry M:V->V such that M(x)=y and we can choose M to be either a reflection or a product of two reflections. (5) (15 points) Let F13 be a finite field with 13 elements. Let V=M3x3(F13), viewed as a vector space over F13, and H(X,Y)=tr(XY) for X,YεV. Show that (V,H) is a quadratic space and find its Witt decomposition. (Notice that 2^2+3^2=0 in F13) (6) (20 points) Let (V,H) be a nondegenerate quadratic space and σ be an isometry of V such that for every anisotropic vector xεV, σx-x is isotropic. Prove that σx-x is isotropic for every xεV. (7) (20 points) Let f(t)=t^3+a2t^2+a1t+a0, g(t)=t^3+b2t^2+b1t+b0 be polynomials with complex coefficients of degree 3, with roots {α1,α2,α3} and {β1,β2,β3} respectively. Show that the resultant Rf,g associated with f(t) and g(t) can be expressed as Rf,g=-(α1-β1)(α1-β2)(α1-β3) (α2-β1)(α2-β2)(α2-β3)(α3-β1)(α3-β2)(α3-β3) (Hint: Find out the homogeneous degree of Rf,g in C[α1,α2,α3,β1,β2,β3] --

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