Composition combines maps by applying one after another: $(S\circ T)(v)=S(T(v))$. For linear maps it is associative, has identity maps as units, and is generally not commutative. Matrix products follow the same order: the rightmost matrix acts first.
Parents
Children
- Definition of composition of linear transformationsFor $T:U\to V$ and $S:V\to W$, define $S\circ T:U\to W$ by $(S\circ T)(u)=S(T(u))$. The intermediate spaces must...
- The composition of linear transformations is a linear transformationIf $T$ and $S$ are linear, then $(S\circ T)(ax+by)=S(aT(x)+bT(y))=aS(T(x))+bS(T(y))$. Thus composition is linear....
- The composition of injective linear transformations is injectiveIf $T$ and $S$ are injective, then $S(T(x))=S(T(y))$ first gives $T(x)=T(y)$ and then $x=y$. Hence $S\circ T$ is...
- The composition of surjective linear transformations is surjectiveIf $T:U\to V$ and $S:V\to W$ are surjective, each $w\in W$ equals $S(v)$ for some $v$, and $v=T(u)$ for some $u$....
- The composition of invertible linear transformations is invertibleFor invertible linear maps $T:U\to V$ and $S:V\to W$, the composition is invertible with $(S\circ T)^{-1}=T^{-1}\circ...
- The inverse of a linear transformation is a linear transformationThe inverse of a bijective linear map is linear. If $y i=T(x i)$, linearity gives $T(ax 1+bx 2)=ay 1+by 2$, so...
- The inverse of the inverse of a linear transformation is the original linear transformationFor invertible $T$, $(T^{-1})^{-1}=T$, because $T\circ T^{-1}$ and $T^{-1}\circ T$ are the appropriate identities....
- The inverse of a composition reverses the orderFor invertible composable maps $A,B$, $(A\circ B)^{-1}=B^{-1}\circ A^{-1}$. Also, for an invertible operator and...
No children match.
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.