Compositions and invertible linear maps

Overview §

Compositions are sequential linear transformations such that the codomain of the first transformation is the domain of the second. The result of taking a composite is also a linear transformation.

Most generally, inverses are linear maps which “undo” another map. An invertible linear map is also called an Isomorphism.

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Compositions of linear maps §

• (Lemma 51) If $A:V\to W$ and $B:W\to U$ are linear maps, then the composite $(BA):V\to U$ defined by $(BA)(v)=B(Av)$ is also linear.
• (Lemma 52) Compositions of linear maps have the following arithmetic properties:
  • (i) If $A:V\to W$, $B:W\to U$, and $C:U\to X$ are linear maps between vector spaces, then $C(BA)=(CB)A$.
  • (ii) If $A:V\to W$ is a linear map, then $A{1}_V=A=1_{W}A$.
  • (iii) If $A:V\to W$ is a linear map and $U$ is any other vector space, then $0_{W,U}A=0_{V,U}$ and $A{0}_{U,V}=0_{U,W}$.

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Inverses §

Definition: Invertible

If $A:V\to W$ is a linear map, we say that

• $A$ is right invertible if there exists a linear map $B:W\to V$ so that $AB=1_W$.
• $A$ is left invertible if there exists a linear map $B:W\to V$ so that $BA=1_V$.
• $A$ is invertible if there exists a linear map $B:W\to V$ so that $AB=1_W$ and $BA=1_V$. We write $B=A^{-1}$.

• (Theorem) Invertible matrix

#wip examples of invertible maps