[線代] 有關可逆的證明
不好意思,我線代總是念得一頭霧水,上來請教高手們一些問題~~
1. let F be a field and A 屬於 Mm×n(F),
B 屬於 Mn×m(F),
if AB=Im BA=In show that m=n
請問該怎麼證呢@@?
2. let F be a field and A,B 屬於Mn×n(F),suppose that AB=In,
show that BA=In and hence A is invertible
我的證明感覺怪怪的,列舉如下
suppose BX=0,then X=(AB)X=A(BX)=0 ∴B is nonsingular
∵AB=In (AB)B^-1=B^-1 A=B^-1 BA=B(B)^-1=In
∵AB=In=BA A is invertible得證
請問這樣證可以嗎@@? 謝謝!!
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