I'm having trouble on a question proving that a function, multiplying a matrix by a column is injective.
The question is:
and
is injective? And why?
Can anyone give me any help?
Let
Therefore:
Suppose two values of are mapped to the same point (ie. is not injective).
Let and let .
This gives:
Since they are mapped to the same point .
Therefore:
Hence:
From the first (and second rows, i'm cutting some working here) .
From the third row .
We want to cancel out the y's.
.
Therefore:
.
We also have: .
Since we already know that , this implies that .
Hence so the function is injective.