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Math Help - B & AB injective but A not injective

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    Member Last_Singularity's Avatar
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    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!
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    Quote Originally Posted by Last_Singularity View Post
    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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