Cancelling Vectors: A=B? Is it Legal?

[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?

 

[quantumdude]

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?

 

[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?

 

[quantumdude]

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

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?

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.