You are given an LT L : R^2 - R^3 with the following properties:
L([1 1]') = [0 sqrt(2) 1]', L([1 − 1]') = [sqrt(2) 0 1]', where ' denotes transpose and sqrt means square root.
Using the natural bases of R2 and R3, find the matrix representation of L.

Question

You are given an LT L : R^2 -> R^3 with the following properties:
L([1 1]') = [0 sqrt(2) 1]', L([1 − 1]') = [sqrt(2) 0 1]', where ' denotes transpose and sqrt means square root.
Using the natural bases of R2 and R3, find the matrix representation of L.

jamesj · Answer

So the natural basis of R^2 is i = (1,0) and j = (0,1)

Hence the vector (1, 1) = i + j and (1, -1) = i - j
and we know the image, L(i + j) and that is equal to L(i) + L(j)

Similarly, we have an expression for L(i-j) = L(i) - L(j)

Therefore you have two simultaneous equations in L(i) and L(j) that you can now solve.

The solution for L(i) and L(j) are the columns of the matrix representing the linear transformation L.