# Math Help - B & AB injective but A not injective

1. ## B & AB injective but A not injective

Let $B: V \rightarrow W$ and $A: W \rightarrow Z$. If their composition $AB$ is injective and $B$ is injective, I know that $A$ does not necessarily need to be injective. Could you please help me find an example of this? Thanks!

2. Originally Posted by Last_Singularity
Let $B: V \rightarrow W$ and $A: W \rightarrow Z$. If their composition $AB$ is injective and $B$ is injective, I know that $A$ does not necessarily need to be injective. Could you please help me find an example of this? Thanks!
$V=Z=\mathbb{R}, \ W=\mathbb{R}^2,$ as vector spaces over $\mathbb{R},$ and define $A,B$ to be the natural projections and injections, i.e. $A(x,y)=x, \ B(x)=(x,0).$