Question

Why are invertible matrices "one-to-one"?

Answer 1

See explanation...

Explanation:

I think the question is referring to the natural use of a matrix to map points to points by multiplication.

Suppose `M` is an invertible matrix with inverse `M^(-1)`

Suppose further that `Mp_1 = Mp_2` for some points `p_1` and `p_2`.

Then multiplying both sides by `M^(-1)` we find:

`p_1 = I p_1 = M^(-1)M p_1 = M^(-1)M p_2 = I p_2 = p_2`

So:

`Mp_1 = Mp_2 => p_1 = p_2`

That is: multiplication by `M` is one-to-one.