Suppose that A is a linear transformation from vector spaces U to V and that B is a linear transformation from vector spaces V to W. Suppose further that B composed of A is an injective composition of the aforementioned linear transformations.
For this composition to be injective, do A and B have to be injective? My intuition tells me that B HAS to be injective while it is not required for A to be injective. Also, how would one prove that either A or B have to be injective.
Also, I am quite fairly new to this site and I have no idea how I can use all of the math symbols and notation that everyone else here is using, so if someone can help me out on that then that would be great.
Best Answer
If $A$ is not injective, there are two elements, $x,y \in U$ such that $x \neq y$, but $A(x) = A(y)$. Then $B$ is stuck, it can't tell $A(x)$ and $A(y)$ apart and sends them both to the same place, so $B(A(x)) = B(A(y))$ and the composition is not injective.
Depending on the details of your notation, it's possible for the image of $A$ to not be all of $V$, so that $B$ could fail to be injective, but $A$ can't send two elements to places that $B$ conflates. I doubt that you are using notation this way; I expect that the image of $A$ is all of $V$ and the image of $B$ is all of $W$. In this case, for the composition to be injective, both $A$ and $B$ must be so:
Therefore, if the composition is injective, so are $A$ and $B$.