I know this is probably a pretty simple thing to do, but I can't really wrap my head around it: I have two vectors in R^2, (1,2) and (1,0), whose transformations are (1,2,0) and (3,0,1) respectively. How do I find out the transformation matrix from this information?
I know I could manually write down a new system of equations using the elements of the matrix as my unknowns, but I supposed that's too tedious to be the right solution.
Also, do basis transformation matrices have anything to do with this? I thought I could use a new matrix consisting of my two column vectors in R2 (representing a change of basis from the standard basis vectors), but I'm not sure if doing the same thing to my transformed vectors in R^3 makes any sense (can I group them up too? Would that be a change of basis in the range?). I'm sorry if I'm mixing up two unrelated things, but I had a hunch the problem might be related to that.
Thanks for the help!
Best Answer
You could try the following. First map the two vectors in R^2 to the standard basis vectors in R^2. Then find a mapping that maps the standard basis vectors in R^2 to the ones in R^3.
In particular, in your example, this would yield something like this: The matrix $A$ that maps the standard basis vectors to (1,2) and (1,0) is:
$$A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} $$
The matrix $B$ that maps (1,0) and (0,1) to (1,2,0) and (3,0,1) is the following one:
$$B = \begin{bmatrix} 1 & 3 \\ 2 & 0 \\ 0 & 1 \end{bmatrix} $$
So, you first want to do the inverse of $A$ to map (1,2) and (0,1) to the standard basis vectors, and then apply $B$ to get to (1,2,0) and (3,0,1). So, the resulting matrix becomes:
$$BA^{-1} = \begin{bmatrix} -5 & 3 \\ 2 & 0 \\ -2 & 1 \end{bmatrix}$$