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︰2007年05月02日
考試時限：120分鐘
是否需發放獎勵金：是
試題 :
￥Suppose B is a basis of the finite dimensional vector space V and
v ∈ V. We use v_B to denote the coordinate of v relative to B.
If T:V → V, we use R_B to denote the matrix representation
of T relative to B. If B' is another basis of V, the change
coordinate matrix C_B,B' is the unique matrix such that
C_B,B' V_B = V_B', for all v ∈ V
(1) Let V = sp{(sinx)^2,(cosx)^2}, B={(sinx)^2,(cosx)^2},v=cos2x.
Find V_B (10pts).
(2) Let V = sp(1,x,x^2,x^3), B =(x^3,x^2,x,1),B'={(x-1)^3,(x-1)^2,
(x-1),1} and v_B = (2,0,0,7). Then v_B' = ? (10pts).
(3) Let V = sp(1,x,x^2), B = (x^2,x,1), and
┌ 5 0 2 ┐
C_B,B' = │ 0 1 1 │
└ 2 0 1 ┘
Find B' (15pts).
(4) Let E be the standard basis of V = R^2 and let T: R^2 → R^2
┌ 1 3 ┐
be the linear transformation with R_E = │ │. Find R_B for
└ 2 4 ┘
B = ([1,1],[1,0]) (15pts).
(5) Is the quadratic form Q(x_1,...,x_99) = x_1^2 + x_2 x_3 + ... +
x_2k x_2k+1 + ... + x_98 x_99 positive definite (7 pts) ? Is there a
maximum or minimum of Q(x_1,...,x_99) (8 pts)? (Why?)
(6) Suppose A is an nxn matrix and v is an eigen-vector of A with
eigen-value λ.
(a): Is λ also an eigen-value of the transpose A^T? Prove it
or give a counter-example (8 pts)
(b): Is v also an eigen-vector of the transpose A^T? Prove it
or give a counter-example (7 pts)
(7) What is the area bounded by the ellipse x^2 + xy + y^2 = 4 ? (10 pts).
(Hint: the area bound by x^2/a^2 + y^2/b^2 = 1 equals abπ ).
(8) Suppose A is an nxn matrix and v_1,...,v_n ∈ R^n are eigen-vectors
of A. And assume that for i≠j, the inner product v_i‧v_j = 0. Is it
nacessary that A^T = A ? Prove it or give a counter-example(10 pts).
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