Linear transformation with two given bases

[aximiliio]

Homework Statement

(a; b) is in terms of D = ( 1,1 ; 1 -1) and (c; d) is in terms of Dx = ( -1,1 ; 0,2), then we need to find a matrix such that
(c;d) = (?, ?; ?, ?)* (a; b).

Homework Equations

y = Ax >> linear transformation

The Attempt at a Solution

I know the answer is [1, -3; 1, -1}...but all I can come up with is that

a(1;1) + b(1;-1) = c(-1;0) + d(1;2)

 

[HallsofIvy]

Please clarify. What vector space are you working in? R² so that (a; b) is the vector $a\vec{i}+ b\vec{j}$? And when you say "(a; b) is in terms of D = ( 1,1 ; 1 -1) and (c; d) is in terms of Dx = ( -1,1 ; 0,2)" do you mean that
1) D is the basis for R² consisting of $\vec{i}+ \vec{j}$ and $\vec{i}- \vec{j}$
2) Dx is the basis for R² consisting of $-\vec{i}+ \vec{j}$ and $2\vec{j}$
3) (a; b) and (c; d) are representations of the same vector in those two different bases?

If those are all true, then the simplest way to find the matrix representing the "change of basis" transformation is to write the two basis vectors in one basis in terms of the other: (1, 1)= a(-1,1)+ b(0,2) for what a, b?

Those vectors (a, b) are the columns of the matrix.