[試題] 107-2 莊武諺 線性代數二 期末考
課程名稱︰線性代數二
課程性質︰數學系大一必修
課程教師︰莊武諺老師
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2019/6/5 (三)
考試時限（分鐘）：150
試題 :
(There are totally 115 points)
(以下ε代表屬於符號)
(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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