projection onto a subspace

(2.3 hours to learn)


The projection of a vector b onto a subspace X is the closest point to b contained in X. Projection is a linear operation, and can be computed using a projection matrix. It is used in linear least squares approximation.


This concept has the prerequisites:

Core resources (read/watch one of the following)


Khan Academy: Linear Algebra
  • Lecture "Projections onto subspaces"
  • Lecture "Visualizing a projection onto a plane"
  • Lecture "A projection onto a subspace is a linear transformation"
  • Lecture "Subspace projection matrix example"
  • Lecture "Another example of a projection matrix"
  • Lecture "Projection is the closest vector in a subspace"
MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
Author: Gilbert Strang


Supplemental resources (the following are optional, but you may find them useful)


See also

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