# Quick question

1. Oct 16, 2005

### tandoorichicken

Do cancelling rules work for vectors? For example, if you had
$$A\vec{x} = B\vec{x}$$
could you cancel out the x's and be left with A = B? Is that legal?

edit: A and B are matrices. Does this make a difference?

2. Oct 16, 2005

### Tom Mattson

Staff Emeritus
Let's fill in some missing steps and then the answer to this question will become clearer.

$$A\vec{x}=B\vec{x}$$
$$A^{-1}A\vec{x}=A^{-1}B\vec{x}$$
$$I\vec{x}=I\vec{x}$$
$$\vec{x}=\vec{x}$$

Now let me ask you: What did I have to assume in order to write each line?

3. Oct 16, 2005

### tandoorichicken

So the assumption was that the matrix A = the matrix B. So then we could take the inverse of each to form the identity matrix on each, and then it followed that Ix = Ix so x=x. So then if we assume that matrices are basically just equivalents of multi-column vectors, and we can cancel them out from both sides, then we should be able to do the same for vectors.

Is that right?

4. Oct 16, 2005

### Tom Mattson

Staff Emeritus
You're right that I had to assume that $A=B$ in order to get $A^{-1}B=I$.

No, there's more to it than that. In the second line I had to assume that $A$ is invertible. And as you would have learned from class, nonsquare matrices are not invertible. So the matrices $A$ and $B$ can't just be any collection of column (or row) vectors.

Note that the case that $A$ is not invertible includes the case that $A=0$ (the zero matrix). In that case the equation $A\vec{x}=B\vec{y}$ is satisfied if $\vec{y}=\vec{0}$ (the zero vector) and $B$ is any matrix of appropriate dimensions. Note that $\vec{x}$ is not necessarily equal to $\vec{y}$ in this case.