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)
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)
C(A)∩N(A):N(A)
C(A)+ N(A):C(A)
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 59.126.0.166