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Homework Statement
Given A is nxn matrix. An=0 and An-1 [tex]\neq[/tex] 0.
Find dim(null (A))
Homework Equations
The Attempt at a Solution
1.
I split An= A An-1=0.
From this, I conclude that
column space of An-1[tex]\subseteq[/tex]null space of A
because if we think that
An-1=[v1 v2 v3 v4 ...] (vi is column vector of An-1),
we can get
An=A An-1 =[Av1 Av2 Av3 ...] = 0.
Then, dim(null(A))[tex]\geq[/tex]dim(Col(A^{n-1})).
2.dim(null(An))=n since An=0.
3.dim(null(An-1)) [tex]\leq[/tex] dim(null(An))
Since if we let v be in null space of An-1, we can get Anv=0.
Then we conclude that v is in null space of An
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