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課程名稱︰線性代數
課程性質︰系必修
課程教師︰陳文進
開課學院：電資學院
開課系所︰資工系
考試日期（年月日）︰2010/1/14
考試時限（分鐘）：180min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(8%)Let a,b,u,v,w€R^4. A = [a u v w], and B = [b u v w] be two 4x4 matrices
, where det(A) = 4 and det(B) = 1. Find det(A+B)
┌0 0 0┐
2.(10%)Let A, B, and P be 3x3 matrices, where B = │1 0 3│ and AP = PB.
└0 1-2┘
Find det(A+I3)
3.(12%)Let V be the plane defined by 2x+y-3z=0.
(a) Find an orthogonal basis for V.
(b) Use (a) to find the distance of the point(2,2,1) to V.
4.(a)(6%)Find the least squares solution of { x1+x2=4
{ 2x1+x2=-2
{ x1-x2=1
(b)(6%)Use (a) to find the orthogonal projection of the point (4,-2,1) to
the plane spanned by vectors [1,2,1]^T and [1,1,-1]^T € R^3
5.(10%)Let P[x]={a+bx|a,b€R} be the vector space consists of polynomials of
degree 1. The inner product of the two functions f(x),g(x)€P[x] is defined
as <f,g> = ∫(0 to 1) f(t)g(t)dt. Find the orthogonal projection of
h(x)=4+3x-x^2 on P[x].
6.(10%)Let A and B be two nxn matrices. Prove that AB and BA have the same
set of eigenvalues.
7.(10%)If a[0]=2, a[1]=3, and a[k+1]=3a[k]-2a[k-1], for all k>=1. Use method
of linear algebra to find the formula of a[k].
8.(a)(8%)Find the standard form of the quadratic curve
3(x^2)+2xy+3(y^2)+6√2 x-10√2 y+10=0.
(b)(4%)Give the coordinates of the center and the vectors representing the
two principal axes of the curve.(the x'-axis must be 0<=θ<π/2 and
the y'-axis must be π/2<=θ<π)
9.(a)(8%)Find the SVD of A = ┌3 2 2┐
└2 3 -2┘
(b)(8%)Use (a) to find an orthogonal basis for each of the four subspaces
R(A), C(a), N(A), and N(A^T).
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心中有變數，就不用變數代換法
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