Linear Transformation Matrix for T: U -> V using Standard Basis of R^2

[mathmathmad]

Homework Statement

T : U -> V is a linear map defined by
T(a,b) = (a-b,a+b)
write down the matrix T using the standard basis of R^2

Homework Equations

The Attempt at a Solution

basis of V = { (1,1) , (-1,1) }

standard basis of R^2 is (1,0) and (0,1)
and the matrix T is essentially ( T(1,0) T(0,1) ) right? where T is a 2x2 matrix
but I don't know how to evaluate T(1,0) and T(0,1)...

 

[HallsofIvy]

Homework Statement

T : U -> V is a linear map defined by
T(a,b) = (a-b,a+b)
write down the matrix T using the standard basis of R^2

Homework Equations

The Attempt at a Solution

basis of V = { (1,1) , (-1,1) }

standard basis of R^2 is (1,0) and (0,1)
and the matrix T is essentially ( T(1,0) T(0,1) ) right?[k/quote]
Yes, that is right.

where T is a 2x2 matrix
but I don't know how to evaluate T(1,0) and T(0,1)...

You are told how in the definition of T!
T(a,b) = (a-b,a+b)

so T(1, 0)= (1-0, 1+0) and T(0, 1)= (0-1, 0+1).

 

[mathmathmad]

oh yeah! oops

say if the basis given are (-1,1) and (0,-2)
what I get for matrix T is

T(-1,1) = (-1-1,-1+1) = (-2,0)
T(0,-2) = (0-(-2),0+(-2) = (2,-2)

so T is (-2, 0
2, -2 )

but when I multiply it with (a,b) to its RHS, I don't get (a-b,a+b)