課程名稱︰工程數學-線性代數
課程性質︰必修
課程教師︰馮世邁
開課學院：電機資訊學院
開課系所︰電機工程學系
考試日期（年月日）︰2018/06/13
考試時限（分鐘）：09:20-10:10（最後延長5分鐘）
試題 :
1. — —
｜ 2 1 1｜
A = ｜ 1 2 1｜
｜ 1 1 2｜
— —
(a)(30%)Find an orthogonal matrz P and a diagonal matrix D such
that A = PD(P^-1)
(b)(10%)Find the spectral decompositions of A and A^2 respectively.
(You may express your answer in terms of vi(vi^T) )
(c)(10%)Find an orthogonal projection matrix for each eigenspace.
(You may express your answer in terms of vi(vi^T) )
2.In the following, determine whether each subset W is a subspace of the
vector space V. (A proof is needed for each answer.)
(a)(10%)Let v be a nonzero vector in R^n. L(R^n,R^m) and W be the set
of all linear transformations T from R^n to R^m such that T(V)=0.
(b)(10%)V=M(n x n) and W is the set of all n x n matrices with rank≦k,
where k is an integer smaller than n.
(c)(10%)V=F(S) and W = {f(t) 屬於 V：f(s1)+f(s2)+...+f(sn)=0}, where
s1,s2,...,sn 屬於 S}
3.(20%)Find the equation of the least-squares line for the data:
(-1,0),(0,0),(1,1).