# Kernel of eigenspace.

1. Nov 30, 2013

### Maths2468

Lets say you have a linear transformation P. The eigenvalues of the matrices are 0,1 and 2.
How would you show that ker P belongs to the eigenspace corresponding to 0?

So you have an eigenvalue 0. Let A be the 3X3 matrix.
I was thinking of doing something like Ax=λx and substitute 0 for λ. And then show that x,y,z are equal to 0 and hence the eigenspace is 0. Would this be a good idea?

2. Nov 30, 2013

### Dick

Thinking a little more about it would be the best idea. Isn't the definition of x being in ker P the same as the definition of x being a eigenvector with eigenvalue 0?

3. Nov 30, 2013

### Maths2468

yes so I assume the original suggestion was bad.

4. Nov 30, 2013

### Dick

It was certainly confusing. If you meant the vector x has the components (x,y,z) then Ax=0 doesn't necessarily mean x,y,z=0.

5. Nov 30, 2013

### Maths2468

Im sorry. I am really really bad/hopeless at linear mathematics. When I meant that A(x,y,z)=0

6. Nov 30, 2013

### Dick

If you are given a specific matrix A and you want to find ker A then that's what you do alright. But you don't have to find ker A to see that it's the same as the set of eigenvectors with eigenvalue 0.

7. Nov 30, 2013

### Maths2468

ok cool. I am starting to understand a little better. How do you know the kernel A is the same as the set of eigenvectors with eigenvalue 0? Where do I go from here?

8. Nov 30, 2013

### vela

Staff Emeritus
What equation does a vector in the kernel of A satisfy? What equation does an eigenvector with $\lambda=0$ satisfy?

9. Nov 30, 2013

### Maths2468

what are you asking for the original matrix?

10. Nov 30, 2013

### vela

Staff Emeritus
No. I'm asking you to tell us how to express the phrase "$\vec{x}$ is in the kernel of a matrix A" mathematically. Similarly, how do you say "$\vec{x}$ is an eigenvector of matrix A with eigenvalue 0" mathematically?

11. Nov 30, 2013

### Maths2468

for "vector x is in the kernel of A" ker(A)={x belongs to X: T(x)=0}
I am not sure about the other one.
Great question by the way, really forcing me to think and understand.

12. Nov 30, 2013

### vela

Staff Emeritus
Your definition for the kernel of A isn't correct as there's no mention of A.

Look up the definition of an eigenvector and eigenvalue. In math, you really should know the definitions.

13. Nov 30, 2013

### Maths2468

What should it be?
to calculate eigenvalue you use Ax=lambda x
I know it is a relatively new topic we have started and I can not stand it. But I try

14. Nov 30, 2013

### Dick

Try to take the think and understand challenge. What's wrong with saying
for "vector x is in the kernel of A" ker(A)={x belongs to X: T(x)=0}? What does T have to do with it? And, yes, x is an eigenvector if Ax=lambda x. What value of lambda are you interested in?

15. Dec 1, 2013

### Maths2468

The value of lambda I am interested in is 0.

AT the end should it be T(A)=0?
Is this stuff needed to answer the question?

16. Dec 1, 2013

### HallsofIvy

No, it isn't! There was no "T" in your question and you cannot just throw one in without defining it! The kernel of A is the set of all x such that A(x)= 0, not "T(x)= 0" as you had before.

Yes, knowing the definition of "Kernel" is needed to answer questions about the "kernel" of a linear transformation!