Thread: Linear operators and finding bases

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.

Originally Posted by Torcida1911

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.

Consider the following 4 vectors in

, , ,

These vectors are linearly independent and span . Thus the vectors form a basis of called the usual basis of .

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.

The standard unit vectors are linearly independent for if we write
The usual or standar basis in component form then we obtain , which implies the coeficients are zeroes. Furthermore the vectors span because an arbitrary vector can be expressed as



We call the usual or standard basis for .

Edit: I think in your problem to find a basis of the kernel of the map T you must set T(v)=0 where v=(x,y,z,t) and then set corresponding components equal to each other to form a homogeneous system whose solution space is the kernel of T.