The linear transformation T: U -> V is defined by T(a,b) = (a-b, a+b). The matrix representation of T using the standard basis of R^2, which consists of (1,0) and (0,1), is derived by evaluating T at these basis vectors. Specifically, T(1,0) results in (1,1) and T(0,1) results in (-1,1), leading to the matrix representation of T as [[1, -1], [1, 1]]. The discussion also explores alternative bases, demonstrating that the transformation's matrix can change based on the chosen basis.
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mathmathmad said: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.
You are told how in the definition of T!where T is a 2x2 matrix
but I don't know how to evaluate T(1,0) and T(0,1)...
T(a,b) = (a-b,a+b)
so T(1, 0)= (1-0, 1+0) and T(0, 1)= (0-1, 0+1).