----- $(\lambda,T)$ denotes the $\lambda$-[[eigenspace]] of a [[linear operator]] $T$. > [!proposition] Proposition. ([[linear projector iff induces direct sum decomposition]]) > Every [[linear projector|linear projector]] $P$ on a finite-dimensional [[vector space]] $V$ induces a [[direct sum of two subspaces|direct sum decomposition]] $V = W \oplus W'=\im P \oplus \ker P = E(1, P) \oplus E(0, P)=\im P \oplus \im (I-P).$ > \ > Conversely, given a [[direct sum of vector spaces]] decomposition $V= W \oplus W'$ there is a [[linear projector]] $P$ such that $\im P = W$, namely $P:V \to V$ given by $Pv = P(v^{(W) } + v^{(W')}) := v^{(W)}.$ > [!proof]- Proof. ([[linear projector iff induces direct sum decomposition]]) > Let $P:V \to V$ be a [[linear projector]] on $V$, then $\ker P \cap \im P = \{ 0 \}$ ([[direct sum of two subspaces|so is a direct sum]]) and by [[dimension of direct sum is sum of dimensions]] and [[Rank-Nullity theorem|Rank-Nullity]] we conclude that $\dim (\ker P \oplus \im P)= \dim V$ that $\ker P \oplus \im P=V$ so $V$ has been decomposed. Because $P$ [[idempotent matrix iff diagonalizable and eigenvalues are all 0 or 1|is diagonalizable with eigenvalues all 0 or 1]] $V = E(0,P) \oplus E(1,P)$ by [[diagonalizable iff V a direct sum of eigenspaces]]. We know that $\im (I-P)=\ker P$ since $(I-P)v = v - Pv = \begin{cases} > 0 & v \in \im P \\ > v & v \in \ker P. > \end{cases}$ > > Conversely, suppose we are given a a [[direct sum of vector spaces]] decomposition $V = W \oplus W'$. Define $P \in$ [[vector space of operators on a vector space]] by $Pv = P(v^{(W) } + v^{(W')}) := v^{(W)},$ > where we can define $P$ in terms of a factorization of the input because said factorization is unique by definition of [[direct sum of vector spaces]]. This is linear because $\begin{align} > P(cv_{1}+v_{2})= & P(cv_{1}^{(W)}+ v_{2}^{(W)} + cv_{1}^{(W')}+ v_{2}^{(W')}) \\ > = & cv_{1}^{(W)} + v_{2}^{(W)} \\ > = & c P(v_{1}^{(W)}) + P(v_{2}^{(W)}). > \end{align}$ > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```