If you know then, the matrix of with respect to the basis is:
you needn't more information about .
Fernando Revilla
B is the basis {v1, v2, v3} of R3 where v1=[1, 1, 1], v2= [2, 3, 0], and v3= [-1, 2, -6] I need to find the matrix A for T with respect to the basis B. But somehow, I can't figure it out. It's given that T: R3--->R3 is a linear transformation and that T(v1)=v2, T(v2)=v3, and T(v3)=v1.
Is T(v1)= [1, 0, 1]? I mean, how do you figure out what T(v) is given the values of v1, v2, and v3? It won't be [(2, 3, 0), (-1, 2, -6), (1, 1, 1)] because that would just be the change of basis right?
If you know then, the matrix of with respect to the basis is:
you needn't more information about .
Fernando Revilla
It is a well known theorem. You only need to transpose the coordinates of on .
Fernando Revilla
You say you are told that "T(v1)= v2". and that v2= [2, 3, 0]. What would make you think that T(v1)= [1, 0, 1]?
What you really need is that T(v1)= v2= 0v1+ 1v2+ 0v3, That T(v2)= v3= 0v1+ 0v2+ 1v3, and that T(v3)= v1= 1v1+ 0v2+ 0v3. You are using [a, b, c] to indicate vectors in the "standard" basis. If you use < a, b, c> to represent vectors in this basis, B, (so that v1= <1, 0, 0>, v2= <0, 1, 0>, and v3= <0, 0, 1>) then you are saying that T(v1)= T(<1, 0, 0>)= <0, 1, 0>, T(v2)= T(<0, 1, 0>)= <0, 0, 1>, and T(v3)= T(<0, 0, 1>)= <1, 0, 0>. What matrix does that?