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Vector Space Projection

ProjectionVectors

A vector space projection onto a subspace W of a vector space V is a linear transformation P:V->W satisfying P(w)=w for every w in W. If W is k-dimensional and V has inner product <,>, an orthogonal projection can be defined as follows. The most familiar projection is when W is the x-axis in the plane. In this case, P(x,y)=(x,0) is the projection. This projection is an orthogonal projection.

If the subspace W has an orthonormal basis {w_1,...,w_k} then

proj_W(v)=sum_(i=1)^k<v,w_i>w_i

is the orthogonal projection onto W. Any vector v in V can be written uniquely as v=v_W+v_(W^_|_), where v_W in W and v_(W^_|_) is in the orthogonal subspace W^_|_.

A projection is always a linear transformation and can be represented by a projection matrix. In addition, for any projection, there is an inner product for which it is an orthogonal projection.

See also

Idempotent, Inner Product, Orthogonal Set, Projection, Projection Matrix, Spectrahedral Shadow, Symmetric Matrix, Vector Space

This entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Vector Space Projection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VectorSpaceProjection.html

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