※ 引述《peropero1 (嗚嗚~~~~)》之銘言：
: 想對個答案跟觀念 麻煩版上大大 謝謝
: €表示 屬於
: Suppose A is an n*n matrix with the property that
: A^2=A.
: Let a1,...,an€R^n, be the column vectors of A and
: A1,A2,...An be€R^n, be the row vectors of A.
: Let C(A)=span(a1,...,an) and R(A)=span(A1,...An) be the column/row space of A
: Define
: E(A)={x€R^n|x=Ax}
: F(A)={x€R^n|x=u-Au for some u€R^n}
: Find the following four sets:
: C(A)∩E(A), N(A)∩F(A),C(A)∩N(A), C(A)+N(A)
: Ans:
: C(A)∩E(A):E(A)
: 對x€C(A)∩E(A)，x€C(A)且x€E(A),
: x€Ax且使得x=Ax, so C(A)∩E(A)=E(A)
這個應該寫C(A) or E(A)都對
: N(A)∩F(A):E(A)
: x為Ax=0且x=u-Au,so u-Au=0 →u=Au, so N(A)∩F(A)=E(A)
for each x€N(A) => Ax = 0 => x-Ax = x => x € F(A)
所以應該是F(A)
: C(A)∩N(A):N(A)
for each x€C(A)∩N(A) =>exists u€R^n Au = x∩ Ax=0
=> Ax = A^2u = Au = x = 0
所以應該是{0}
: C(A)+ N(A):C(A)
因為dim(C(A)∩N(A)) = 0 => C(A)+N(A) = R^n
有錯請指教!!
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