Projections onto Subspaces and Least Squares projection matrix
In this lecture, we learn about projections onto subspaces and how to compute them using linear algebra. The focus is on projecting a vector B onto a one-dimensional subspace described by a vector A and onto a two-dimensional subspace described by two vectors A1 and A2. The lecture also covers the concept of the projection matrix, which acts on the input vector B and produces the projection P. The formula for the projection matrix is derived and its properties are discussed, including its symmetry and the fact that it is a rank one matrix. The lecture also explains the importance of projections in solving systems of equations and finding the best approximation to a vector in a subspace.
Projections onto Subspaces and Least Squares projection matrix -- Lecture 15. Learn what a projection is in Linear Algebra and how to compute a projection.
Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 22, 2008). License: Creative Commons BY-NC-SA.
More info at: http://ocw.mit.edu/terms
- Least Squares Java Applet - Java applet to play around with to investigate Least Squares lines / equations.
- Least Squares Java Applet 2 - Another interactive Least Squares java applet to play around with.
- What does it mean to project a vector onto another vector?
- How do you compute a projection matrix?
- Why do you project a vector or matrix?
-
Staff Review
This lesson explains what a projection is and how to find the projection vector. A very important lesson in the grand scheme of Linear Algebra, and very well-taught.
