Linear Algebra- Kernel and images of a matrix

[KyleS4562]

Homework Statement

Consider a square matrix A:

a. What is the relationship between ker(A) and ker(A^2)? Are they necessarily equal? Is one of them necessarily contained in the other? More generally, What can you say about ker(A), ker(A^2), ker(A^3), ker(A^4),...?

b. What can you say about im(A), im(A^2), im(A^3), im(A^4),...?


2. The attempt at a solution

So i believe if A is invertible nxn matrix, than ker(A)={<0,0,0>} and so will ker(A^2) and so on. And the image of A if A is invertible is im(A)=R^n and so will the im(A^2) and so on, but I am not sure what it would be for other conditions of A at least that's what I think this question wants.

 

[Donaldos]

a. If x \in \ker(A), what can you say about A^2 x? What does that tell you about \ker(A^2)?

b. For any x, A^2x is in \operatorname{im}(A^2). If you rewrite A^2x as A(Ax)=Ay, what can you say about \operatorname{im}(A)?

 

[KyleS4562]

for A: A^2 * x would equal the zero vector which means the ker(A^2) contains x but is not necessarily equal because it may contain another vector

B. Is that saying the im(A) contains the im(A^2) but is not necessarily equal to the im(A^2) because it will span more vectors?

 

[Donaldos]

for A: A^2 * x would equal the zero vector which means the ker(A^2) contains x but is not necessarily equal because it may contain another vector

In other words, we have \ker(A)\subset \ker(A^2).

B. Is that saying the im(A) contains the im(A^2) but is not necessarily equal to the im(A^2) because it will span more vectors?

Yes. You could also write : \operatorname{im}(A^2)\subset \operatorname{im}(A)

Now, how can you apply this to \ker(A^3), \ker(A^4)... and \operatorname{im}(A^3), \operatorname{im}(A^4)...?

 

[KyleS4562]

alright, thank you very much for your help