B & AB injective but A not injective

**Last_Singularity** · September 24th 2009, 09:06 PM

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!

**NonCommAlg** · September 24th 2009, 10:33 PM

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. $A(x,y)=x, \ B(x)=(x,0).$

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B & AB injective but A not injective

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