The relation between span(In,A,A2, )and it's minimal polynomial

[alazhumizhu]

Let A ∈ Mn×n(F )
Why dim span(In, A, A2, A3, . . .) = deg(mA)?? where mA is the minimal polynomial of A.
For span (In,A,A2...)

I can prove its

dimension <= n by CH Theorem

but what's the relation between

dim span(In,A,A2...)and deg(mA)

 

[micromass]

For example, if the minimal polynomial is $x^2+x+1$. Then $A^2+A+1=0$.

Do you see any way to conclude that $A^2\in span(A,1)$?

 

[alazhumizhu]

ybut i don't know why can't I write A

 

[alazhumizhu]

ybut i don't know why can't I write A in spanA2I

 

[alazhumizhu]

y,but i don't know why can't I write A in span{A2,I}?
I'm sorry about the type..

 

[micromass]

y,but i don't know why can't I write A in span{A2,I}?
I'm sorry about the type..

That's also true, but I don't see how this fact helps you solve the problem.