Linear Algebra- Kernel and images of a matrix

  • Thread starter KyleS4562
  • Start date
  • #1
18
0

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 im not sure what it would be for other conditions of A at least thats what I think this question wants.
 

Answers and Replies

  • #2
68
0
a. If [tex]x \in \ker(A)[/tex], what can you say about [tex]A^2 x[/tex]? What does that tell you about [tex]\ker(A^2)[/tex]?

b. For any [tex]x[/tex], [tex]A^2x[/tex] is in [tex]\operatorname{im}(A^2)[/tex]. If you rewrite [tex]A^2x[/tex] as [tex]A(Ax)=Ay[/tex], what can you say about [tex]\operatorname{im}(A)[/tex]?
 
  • #3
18
0
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?
 
  • #4
68
0
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 [tex]\ker(A)\subset \ker(A^2)[/tex].

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 : [tex]\operatorname{im}(A^2)\subset \operatorname{im}(A)[/tex]

Now, how can you apply this to [tex]\ker(A^3)[/tex], [tex]\ker(A^4)[/tex]... and [tex]\operatorname{im}(A^3)[/tex], [tex]\operatorname{im}(A^4)[/tex]...?
 
  • #5
18
0
alright, thank you very much for your help
 

Related Threads on Linear Algebra- Kernel and images of a matrix

  • Last Post
Replies
4
Views
2K
Replies
5
Views
2K
Replies
1
Views
7K
Replies
19
Views
5K
Replies
1
Views
2K
  • Last Post
Replies
3
Views
5K
  • Last Post
2
Replies
36
Views
2K
  • Last Post
Replies
9
Views
1K
M
Replies
2
Views
7K
  • Last Post
Replies
4
Views
28K
Top