 Projection (linear algebra)

A linear projection is an idempotent linear operator: $P^2=P$. Every vector decomposes as $x=Px+(I-P)x$, into the image of a linear map and the kernel of a linear map. Those two vector subspaces form a direct sum. An orthogonal projection additionally uses the orthogonal complement in an inner-product space.

In linear algebra, **projection** refers to the operation of mapping a vector onto a subspace. The result of this operation is the closest vector in the subspace to the original vector. This concept is crucial in various applications such as computer graphics, machine learning, and statistics. ### Key Concepts 1. **Subspace**: A subspace is a vector space that is part of a larger vector space.