Vector Space Projection — from Wolfram MathWorld
Excerpt
If W is a k-dimensional subspace of a vector space V with inner product <,>, then it is possible to project vectors from V to W. 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 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^|),…
TOPICS
If 
is a 
-dimensional subspace of a vector space 
with inner product 
, then it is possible to project vectors from 
to 
. The most familiar projection is when 
is the x-axis in the plane. In this case, 
is the projection. This projection is an orthogonal projection.
If the subspace 
has an orthonormal basis 
then
is the orthogonal projection onto 
. Any vector 
can be written uniquely as 
, where 
and 
is in the orthogonal subspace 
.
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, Projection Matrix, Orthogonal Set, Projection, Symmetric Matrix, Vector Space
This entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd. “Vector Space Projection.” From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/VectorSpaceProjection.html