課程名稱︰線性代數
課程性質︰資工系大二上必修
課程教師︰李明穗
開課學院：電機資訊學院
開課系所︰資訊系
考試日期（年月日）︰97.11.13
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Problem 1 (20 points)
(a)(10 points) Let Ａ屬於Ｆ(m*m) , Ｃ屬於Ｆ(n*n) be invertible matrices.
┌ Ａ Ｂ ┐
Please prove that Ｘ = │ │ is an invertible matrix(where B屬於
└ Ｏ Ｃ ┘
Ｆ(m*n) and Ｏ is a zero matrix).
┌ Ｐ Ｑ ┐
(Hint: You can let Ｙ = │ │ and solve for Ｙ in ＸＹ=Ｉ, where I
└ Ｒ Ｓ ┘
is an identity matrix.)
┌ 3 1 2 6 5 ┐
│ 1 2 0 4 3 │
(b)(10 points) Find the inverse matrix of Ａ = │ 1 0 1 2 1 │
│ 0 0 0 4 1 │
└ 0 0 0 3 2 ┘
(Hint: You can solve oit easily if you use ;the result of (a).)
Problem 2 (15 points)
(a)(5 points) Prove that sinx, sin2x, sin3x are linearly independent.
(b)(5 points) Prove that if A={x1,x2,x3} is a basis of a vector space V,
so is B={x1,x1+x2,x1+x2+x3}
(c)(5 points) a,b屬於實數, X={x1,x2,x3} = {(1,2,0),(2,1,0),(a,b,1)},
if {x1,x2,x3} is a basis of 實數^3, please find all possible pairs(a,b).
Problem 3 (15 points)
X1 + X3 = 2
2*X1 + 2*X2 + + 3*X4 = 1
4*X2 - 4*X3 + 5*X4 = -7
(a)(9 points) Please find an LU decomposition of the coefficient matrix Ａ;
(b)(6 points) Find the solutions(ifany) for the system
Problem 4 (25 points)
┌ 1 2 5 0 3 ┐
Let Ａ = │ 0 1 3 0 0 │
│ 0 0 0 1 1 │
└ 0 0 0 0 0 ┘
(a)(5 points) Find the row space of Ａ;
(b)(5 points) Find the column space of Ａ;
(c)(5 points) Find the null space of Ａ;
(d)(5 points) Find rank(Ａ) and nullity(Ａ);
(e)(5 points) Is the space got in (a) orthogonal to the space got in (c)?
If your answer is true, prove if. If false, give a counterexample.
Problem 5 (25 points)
Which of the following T are linear transformations? If it is a linear
transformation, prove it, Otherwise, give a counterexample of explain
the the reason.
(a)(5 points) T(x,y,z) = (x-y,x^2,2z)
(b)(5 points) T(x,y,z) = (2x-3y,3y-2z,2z)
(c)(5 points) T(x,y) = (x-y,2x+2)
(d)(5 points) Let P^2 = { at^2 + bt + c | a,b,c屬於Ｒ }, where Ｒ is the set
of real number. The transformation is defined as:
T: P^2 → P^2 by T( a2*t^2 + a1*t + a0 ) = (a2-1)t^2
(e)(5 points) Let V be the space of all 2x2 matrices with real entries.
Let T: V → V be defined by T(Ａ) = Ａ* where Ａ* is the transpose
matrix of Ａ.
Problem 6 (15 points)
V = 複數(n*n), F=複數, for all A,B,C 屬於 V
Define <A,B> = tr(AB^H) (ps: tr(AB^H) means the sum of diagonal elements of
matrix(AB^H))
Prove that <,> is an inner product on V.
(ps. B^H is the Hermitian matrix of B which is a matrix with complex entries.
┌ 2+i 3i ┐
For example, if B = │ 4-i 5 │, the B^H = ┌ 2-i 4+i 0 ┐)
└ 0 0 ┘ └ -3i 5 0 ┘
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