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

1. Jan 27, 2013

### 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)

2. Jan 27, 2013

### micromass

Staff Emeritus
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 $A^2\in span(A,1)$?

3. Jan 27, 2013

### alazhumizhu

ybut i dont know why cant I write A

4. Jan 27, 2013

### alazhumizhu

ybut i dont know why cant I write A in spanA2I

5. Jan 27, 2013

### alazhumizhu

y,but i dont know why cant I write A in span{A2,I}?