Proving Ker P ∈ Eigenspace for Eigenvalue 0 in Linear Transformation

[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?

 

[Dick]

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?

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?

 

[Maths2468]

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?

yes so I assume the original suggestion was bad.

 

[Dick]

yes so I assume the original suggestion was bad.

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.

 

[Maths2468]

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.

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

 

[Dick]

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

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.

 

[Maths2468]

If you are given a 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.

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?

 

[vela]

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

 

[Maths2468]

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

what are you asking for the original matrix?

 

[vela]

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?

 

[Maths2468]

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?

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.

 

[vela]

for "vector x is in the kernel of A" ker(A)={x belongs to X: T(x)=0}

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

I am not sure about the other one.

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

 

[Maths2468]

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.

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

 

[Dick]

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

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?

 

[Maths2468]

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?

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?

 

[HallsofIvy]

The value of lambda I am interested in is 0.

AT the end should it be T(A)=0?

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.

Is this stuff needed to answer the question?

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