Linear Transformation Homework: [T]BC, [v]B, and T(v) Explained

[Maxwhale]

Homework Statement

T: R3 --> R2 by T(x,y,z) = (z-x , 2y -x)
v = (2, -1, -3)
B = {(0,0,1),(0,1,1),(1,1,1,)}
C = {(1,-1), (2,1)}

What is [T]_BC
what is [v]_B
and what is T(v)

Homework Equations

No clue

The Attempt at a Solution

I found out [T]_B and that's where i am stuck.

 

[Mark44]

I'm pretty sure that $$[T]_{BC}$$ is the matrix representation of T on the vectors in B, in terms of the basis vectors in C.

For example, T(b1) = T(0,0,1) = (1,0)
In terms of the basis in C, this is 1/3 (1, -1) + 1(2,1)

So the first column of $$[T]_{BC}$$ is (1/3, 1) (written as a column vector).
You need to do this for the other two vectors in B.

$$v_B$$ is the representation of v in terms of the basis B.
T(v) is pretty straightforward.