Linear transformation with 2 ordered basis

[squaremeplz]

Homework Statement

Suppose L:R^2 -> R^3

Find the matrix representing L(x) = Tx with respect to the ordered basis [u1,u2] and [b1.b2,b3]

Homework Equations

The Attempt at a Solution

I've excluded the actual values since i can do the computation. Just wanted to make sure these steps are ok and I should get 2 different matrices (one for each base)

1. [x]_u = u^(-1) [x]_e

[T(x)]_u = u^(-1) Ax

2. [x]_b = b^(-1) [x]_e

[T(x)]_b = b^(-1) Ax

then L with respect to u is

[T(x)]_u= [u^(-1)]*T*u

and L with respect to b is

[T(x)]_b = [b^(-1)]*T*b

(T) is the transformation matrix with respect to standard basis e
sorry for the poor notation.

much appreciated!

 

[HallsofIvy]

I'm not sure what kind of answer you want. Of course, a specific matrix would depend upon the specific T. The standard way to find the matrix representation of a linear transformation is to apply the linear transformation to each vector in the first basis, here {u1, u2} and write the result as a linear combination of vectors in the second basis, here, {v1, v2, b3}.

I don't know what you mean by "(T) is the transformation matrix with respect to standard basis e". The standard basis of which vector space?