For $T:V\to W$, the main associated subspaces are $\ker T\subseteq V$ and $\operatorname{range}T\subseteq W$. For an operator $V\to V$, also study eigenspaces $\ker(T-\lambda I)$ and stabilized kernels/images of powers. Inner products add orthogonal complements; these should not be confused with arbitrary algebraic complements.
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- The range/image of a linear transformation is a subspace.The image of a linear map is a subspace of its codomain. For images $u=T(x)$ and $v=T(y)$, $au+bv=T(ax+by)$ is again...
- The inverse image of a subspace under a linear transformation is a subspace.If $U$ is a subspace of the codomain of a linear map $T$, then $T^{-1}(U)$ is a subspace of its domain. It contains...
- Definition of kernel/null space of linear transformationThe kernel or null space of $T:V\to W$ is $\ker T=\{v\in V:T(v)=0 W\}$. It measures lost directions. A linear map is...
- The kernel/null space of a linear transformation is a subspaceThe kernel contains zero and is closed under combinations: if $T(u)=T(v)=0$, then $T(au+bv)=0$. Hence it is a...
- Definition of generalized kernel/null space of linear transformationFor $T:V\to V$ with $\dim V=n$, the generalized kernel is the stabilized subspace $\ker(T^n)$. It contains vectors...
- Finite-dimensional Fitting decomposition into generalized kernel and rangeFor $T:V\to V$ on an $n$-dimensional space, $V=\ker(T^n)\oplus\operatorname{range}(T^n)$ (Fitting decomposition). If...
- The kernels of powers of a linear transformation form an ascending chainFor any operator, $\ker T^k\subseteq\ker T^{k+1}$ because $T^kx=0$ implies $T^{k+1}x=0$. In dimension $n$ the chain...
- The range spaces of powers of a linear transformation form a descending chainFor an operator, $\operatorname{range}T^{k+1}\subseteq\operatorname{range}T^k$, because $T^{k+1}x=T^k(Tx)$. In finite...
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