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04月04日
考試時限：120分鐘
是否需發放獎勵金：是
試題 :
(1) Transforn the basis [0,-2,1,2],[2,2,1,6],[0,4,0,4] of R^4 into an
orthogonal basis, using the Gram-Schmidt process (25 points). And
find the projection of the vector [1,1,1,1] onto the space
W= sp([0,-2,1,2],[2,2,1,6]) (10 points)
(2) We have a set of data:(x1,y1)=(-1,2),(x2,y2)=(0,5),(x3,y3)=
(1,9). Explain the least-square method and use it to find the linear fit
y=Ax+B (25 points)
(3) Suppose that A is an nxn orthogonal matrix which has only real
eigen-values.
(a) Show that an eigen-value of A is either 1 or -1. (10 points)
(b) If v,w ∈ R^n are eigen-vectors of A of different eigen-values,then
v‧w = 0 (5 points)
(4) Let T:R^n —→R^n be an linear transformation. Show that T is
orthogonal if and only if T maps unit vector to unit vectors. (10 points)
(5) Find a 4x4 projection matrix of rank 3 such that none of its entries
is zero. (10 points)
(6) Let A be an nxn and let A^T be its transpose. Show that for
v,w ∈ R^n, we have
v‧Aw=A^Tv‧w (5 points)
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