MathGloss

A projection is a linear transformation $P:V\to W$ between vector spaces $V$ and $W$ such that $P^2 = P$. That is, $P$ is idempotent when considered as an element of the ring of linear transformations.

Equivalently, for $V’$ a subspace of a vector space $V$, a linear transformation $P:V\to V$ is a projection onto $V’$ if

1. $P_{\vert V’} = \text{id}_V’$. That is, if the restriction of $P$ to $V’$ is equal to the identity function on $V’$, and
2. $\text{im}(P) \subseteq V’$.

Wikidata ID: Q519967

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