課程名稱︰線性代數一
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︰2006年11月01日
考試時限：未知
是否需發放獎勵金：是
試題 :
Write your answer on the answer sheet. You should include in your
answer every piece of computations and every piece of reasonings so
that corresponding partial credit could be gained.
In this exam, let
┌ 1 2 -3┐ ┌0 1 4 1┐ ┌1 1┐
A = ｜ 3 5 2｜ , B =｜6 2 1 3｜ and C = └0 1┘ .
└-2 -3 -4┘ ｜6 1 -3 2｜
└3 1 4 0┘
(1)Find A^-1(A inverse) and find X such that
┌11┐
AX = ｜ 1｜ (30 points)
└ 6┘ .
(2)Find bases of the column space, the row space and the nullspace
of B. (30 points)
(3)Let T : R^2 → R^2 be the linear transformation so that C is the
matrix corresponding to T. For a set C ⊂R^2, let
T(C)={Y∈R^2 | Y=T(X) for some X ∈ C}.
(a)Show that if V⊂R^2 is a linear subspace, then so is T(V).
(10 points)
(b)Show that if L⊂R^2 is a straight line, then T(L)
is also a straight line. For the straight line L0 defined by
the equation x+2y=3, find the equation for T(L0). (10 points)
(4) (a)Find a square matrix D such that D^3=0 but D^2≠0. (5 points)
(b)Find a square matrix E such that E^3=I but E^2≠I. (5 points)
(5)Suppose that F and G are n*n square matrices. Prove that F and G
are invertible if and only if so is FG.
試題結束.
P.S.標成淺藍色的字是試題卷上的特殊字體
其中淺藍的C和白色的C是不同的(白色C是指上面的C矩陣)
若有錯誤還請幫忙指出^^"
另外......沒有解答,不好意思...orz
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