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）︰2018/06/27
考試時限（分鐘）：09:20-11:00
試題 :
Use of all automatic conputing machines including calculator is prohibited
1. — —
｜ 1 -2 2｜
A = ｜-4 3 -4｜
｜-8 8 -9｜
— —
(a)( 6%) Find the characteristic polynomail of A.
(b)(14%) Find an invertible matrix P and a diagonal matrix D such that
A = PD(P^-1).
2.Let W = Span{u1, u2, u3}, where
— — — — — —
｜ 1｜ ｜ 2｜ ｜ 2｜
u1 = ｜ 0｜, u2 = ｜ 1｜, u3 = ｜-1｜
｜-1｜ ｜-1｜ ｜-1｜
｜ 1｜ ｜ 0｜ ｜ 3｜
— — — — — —
(a)(14%)Find an orthonormal basis {w1, w2, w3} for W.
(b)( 6%)Let v = [1 1 1 1]^T. Find a vector w in W and a vector z in
W^┴ such that v = w + z.
3.(10%)Let W = Null[1 1 1 1]. Find the orthhogonal projection matrix Pw.
4.(10%)Find two 4 x 1 vectors q3 and q4 auch that Q = [q1 q2 q3 q4] is
orthogonal, where q1 and q2 are respectively given by
— — — —
｜ 1｜ ｜ 1｜
q1 = 0.5 ｜ 1｜, q2 = 0.5 ｜ 1｜
｜ 1｜ ｜-1｜
｜ 1｜ ｜-1｜
— — — —
— —
｜ 1 -1 ｜
5.Let B = ｜ 1 1 ｜. Let W = {A 屬於 M(2x2) ： AB = BA }
— —
(a)(10%)Prove that W is a subspace of M(2x2).
(b)(10%)Find a basis for W.
6.Let T : P2 → P2 be a linear transformation defined by
T(p(x)) = p(0) + 3p'(1)x + p(2)x^2, where p'(x) is the derivative of p(x).
Let B = { 1, x, x^2} be a basis for P2.
(a)( 8%)Find the matrix representation of T with respect to B, i.e.,
find [T]B.
(b)(12%)Find a basis for the null space of T and find a basis for the
range of T.