課程名稱︰線性代數二
課程性質︰數學系必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期︰2011年03月25日
考試時限：110分鐘
是否需發放獎勵金：是
試題 :
LINEAR ALGEBRA EXAM I
3/25 2011
Write your answer on the answer sheet.
In this examination, C([-1,1]) denote the inner product space consisting of
all real valued continuous functions on the close interval [-1,1] with the
inner product defined by 1
<f,g> = ∫f(x)g(x)dx.
-1
Also, P_k(F) denote the vector formed by all polynomials of degree not greater
than k with coefficients in F.
┌1 -4 -3┐
(1) (25 points) Let A =│0 -1 -1│. Calculate the characteristic polynomial of
└0 0 0┘
A (10 points), find all the eigenvalues of A (5 points) and find a matrix
Q so that QAQ^-1 is diagonal (10 points).
(2) (20 points) Let S =｛v_1,v_2,v_3｝be the ordered basis of standard inner
product space R^3, with v_1 = (2,2,1), v_2 = (1,0,1), v_3 = (0,1,2). Apply
the Gram-Schmidt process to S to obtain an orthogonal basis of R^3 (10
points). Then normalize the vectors in this basis to obatin an orthonormal
basis β＝｛u_1,u_2,u_3｝and find a_1,a_2,a_3 so that (3,2,5) = (a_1)(u_1)
+ (a_2)(u_2) + (a_3)(u_3) (10 points).
(3) Either give a brief reason or give a counter example for each of the foll-
owing assertions (5 points each):
(a) We have (S⊥)⊥ = Span(S), for each subset S of a finite dimensional
inner product space.
(b) Suppose f(x) ∈ C([-1,1]) has the nth derivative f(n)(x) and T_n(x) =
n f(k)(0)
Σ────x^k is its nth Taylor polynomial. Then ||f-T_n|| ≦ ||f-g||
k=0 k!
for all g ∈ P_n(R).
(c) An inner product space V is spanned by the subset
｛v∈V | <v-v_0,v-v_0> ≦ ε｝,
for any given positive number ε and any given vector v_0 ∈ V.
(d) Every matrix A satisfying A^2 = 0 is diagonalizable.
(e) If a 2x2 matrix A satisfies A^2 = I, then its characteristic polynomi-
al equals x^2 - 1.
(4) Give a rigorous proof for each of the following assertions.
(a) (15 points) Suppose T and U are diagonalizable linear operators on a
finite dimensional vector space V. Show that T and U are simultaneous-
ly diagonalizable if and only if T。U = U。T.
(b) (10 points) Let V = C([-1,1]). Suppose W_e and W_o denote the subspac-
es of V consisting of the even and odd functions respectively. Prove
that W_e⊥ = W_o.
(c) (5 points) In an inner product space with order basis｛v_1,...,v_n｝,
the matrix A = (a_ij), with the ij entry a_ij = <v_i,v_j>, is inverti-
ble.
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