[理工] 線代 零核空間

[sandy900113 (yoyobear)]

http://i.imgur.com/YUiyki7.jpg
想請問一下這題b選項的null(A)為什麼也會是x1,x2的展延呢?

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→ Gabino: 是正交補空間
→ yupog2003: ∀x ∈N(A) => Ax=0 => A^tAx=A^t0=0 => x∈N(A^tA)
→ yupog2003: ∀x ∈N(A^tA) => A^tAx=0 => x^tA^tAx=x^t0=0 =>
→ yupog2003: (Ax)^t(Ax)=0 => <Ax,Ax>=0 => ||(Ax)||^2=0 => Ax=0
→ YuxiWen: 因為ker(AtA)=ker(A)
→ yupog2003: =>x ∈N(A)
→ yupog2003: 由以上兩式得到ker(A^tA)⊆ker(A)且ker(A)⊆ker(A^tA)
→ yupog2003: =>ker(A)=ker(A^tA)
→ sandy900113: 懂了  感謝!!