Range of a linear transformation to power n

[Lostmant]

Homework Statement

How to prove that: the range of a square matrix A (linear transformation) to the power of n+1 is a subspace of the Range of A to the power n, for all n >= 1?

i.e. Range (A^(n+1)) is a subspace of Range (A^n)

Homework Equations

The Attempt at a Solution

I can prove that Kernal (A^n) is a subspace of Kernal (A^(n+1)). Not sure if this is the basis of the prove.

 

[jbunniii]

Hint:

$$A^{n+1}x = A^n(Ax)$$

 

[Lostmant]

Thank you jbunniii. Using the hint you gave me I could prove that Kernal (A^n) is a subspace of Kernal (A^(n+1)), because A^n(A(0)) = 0. but what about range? I think if i could prove the range of A is always a subspace of domain of A, then i am done. How to prove this? Thanks.

 

[jbunniii]

If y is in the range of $A^{n+1}$, then by definition,

$$y = A^{n+1}x$$

for some x.

But

$$y = A^n(Ax)$$

So what does that imply?

 

[Lostmant]

Got it. Thanks so much.