# How to find the transformation matrix given two vectors and their particular transformations?

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 $$\mathbb{R}^2$$, $$〈1,2〉$$ 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 $$\mathbb{R}^2$$ (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 $$\mathbb{R}^3$$ 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!

You could try the following. First map the two vectors in $$\mathbb{R}^2$$ to the standard basis vectors in $$\mathbb{R}^2$$. Then find a mapping that maps the standard basis vectors in $$\mathbb{R}^2$$ to the ones in $$\mathbb{R}^3$$.

In particular, in your example, this would yield something like this: The matrix $$A$$ that maps the standard basis vectors to $$\langle 1,2\rangle$$ and $$\langle 1,0\rangle$$ is:

$$A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$$

The matrix $$B$$ that maps $$\langle 1,0\rangle$$ and $$\langle 0,1\rangle$$ to $$\langle 1,2,0\rangle$$ and $$\langle 3,0,1\rangle$$ 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 $$\langle 1,2\rangle$$ and $$\langle 0,1\rangle$$ to the standard basis vectors, and then apply $$B$$ to get to $$\langle 1,2,0\rangle$$ and $$\langle 3,0,1\rangle$$. So, the resulting matrix becomes:

$$BA^{-1} = \begin{bmatrix} -5 & 3 \\ 2 & 0 \\ -2 & 1 \end{bmatrix}$$

• Ah, I see, that's sort of the kind of reasoning I tried to do. Though I still don't quite understand what B is supposed to be. Would it be a change of basis matrix too, just as A is? Or is it just a transformation matrix for the range? Thanks! Commented Sep 16, 2016 at 20:48
• It's probably best to picture $B$ as a function that takes vectors from $\mathbb{R}^2$, and maps them to $\mathbb{R}^3$. As you are mapping from one vector space to a different one, this is not a basis transformation. I'm not quite sure what you mean by a "transformation matrix for the range". :) Commented Sep 16, 2016 at 20:51
• Heh, by that expression I meant to say what you ended up saying: some sort of function that gives you the transformation vector. One last thing, if you don't mind: why can't I use this function B as my transformation matrix? Is it because it's in a non-standard basis (i.e. generated by (1,2) and (1,0))? Commented Sep 16, 2016 at 21:16

HINT....You can write this as a matrix equation. Let the transformation matrix be $M$

Then $$M\left(\begin {matrix}1&1\\2&0\end{matrix}\right)=\left(\begin{matrix}1.2&3\\0&0.1\end{matrix}\right)$$

Now post multiply by the inverse and calculate $$M=\left(\begin{matrix}1.2&3\\0&0.1\end{matrix}\right)\left(\begin {matrix}1&1\\2&0\end{matrix}\right)^{-1}$$

Now you can finish this.

• Unfortunately this doesn't seem to be right, the question talks about vectors in $\mathbb{R}^2$, this answer instead changes them to vectors in $\mathbb{R}^2$. Commented Sep 30, 2018 at 14:05