課程名稱︰ 線性代數
課程性質︰ 資訊系系必修
課程教師︰ 陳文進
開課學院： 電資學院
開課系所︰ 資訊工程學系
考試日期（年月日）︰ 1.15.2008
考試時限（分鐘）： 14:30~17:20
是否需發放獎勵金：yes
（如未明確表示，則不予發放）
試題 :
1.(15%) f(t)=t-1 and g(t)=t^2+t are two functions in Ρ2 ○ρ([0,1])
1
and the inner product of f(t) and g(t) is ∫f(t)g(t)dt. Find the othogonal
0
complement of span(f,g).
2.(10%) Let V ○R^3 be the subspace defined by
V = {(x1,x2,x3): x1 - x2 + x3 = 0}.
Find the standard matrix for each of the following linear transformations.
(i) Projection on V.
(ii) Reflection across V.
3.(15%) Let A=[aij] be an n*n matrix where aij= -aji for all 1≦i,j≦n
and n be an odd number. Find the dterminant of A.
4.(15%) Letλ be an eigenvalue of an n*n matrix A.
(i) Find the eigenvalues of A^2 and A^2 + 2A + 3I, where I is the n*n
identity matrix.
(ii) Find the eigenvalues of A^(-1) and I - A^(-1).
5.(15%) A is a 3*3 matrix and χ, Aχ, A^2χ are three linear independent
vectors where A^3χ = 3Aχ - 2A^2χ. Let P be the 3*3 matrix where χ, Aχ,
and A^2χ are the first, second, and third columns of P.
(i) Find the 3*3 matrix B such that P^(-1)AP = B.
(ii) Find the determinant of (A + I).
6.(15%) Find the standard form of the following quadratic curve.
5x1^2 - 6x1x2 + 5x2^2 - 4x1 - 4x2 - 4 = 0
Please give also the new origin and the equations of the two new coordinate
axes in the original coordinate system that makes the curve has the standard
form.(意即給出在原座標中，曲線的中心座標與兩對稱軸所在直線的方程式)
7.(15%) If a0 = 0, a1 = a2 = 1, and a(k+1) = 2ak + a(k-1) - 2a(k-2) for k≧2,
use methods of linear algebra to determine the formula for ak.
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