Hi, I'm having trouble with a couple of questions.

The quesiton reads

A linear operator T on R^4 has the matrix

A= [some 4x4 matrix]

with respect to bases Alpha=Beta=E4.

Find the basis for T(R^4).

-I understand what to do when the question reads find T(1,2,-4,0) or something like that, I'm not sure what to do when it asks me to find a basis for all of R^4.

If someone could please explain the steps on how to tackle this problem I would be appreciative of that.

Secondly, the question later reads

Find the basis of the kernel of T

and

Find the standard basis of the kernel of T.

If anyone could clarify what is the difference between the two and how the calculation procedures are different I would also appreciate that.

Thank you very much.