Orthogonal Matrices and Gram-Schmidt
In this lesson, we learn about orthogonal matrices and Gram-Schmidt. An orthonormal vector and basis are introduced, and we see how having an orthogonal basis can make calculations better. Working with orthonormal vectors is useful in numerical linear algebra, as they never get out of hand. The lecture then shows how to make a given basis into an orthonormal one using the Gram-Schmidt process. Projection matrices are also discussed, and we learn that if we have an orthonormal basis, the projection matrix becomes the identity matrix. All the equations of this chapter become trivial when we have an orthonormal basis.
Orthogonal Matrices and bases, and Gram-Schmidt -- Lecture 17. Learn what an orthonormal vector and basis looks like and why it is convenient.
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
- Gram-Schmidt Java Applet - Java applet to play around to investigate Gram-Schmidt and Orthogonal matrix operations.
- What is an orthogonal basis?
- What is an orthogonal matrix?
- What is Gram-Schmidt?
- What are orthonormal vectors?
- What is an orthonormal basis?
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Staff Review
The final lecture on orthogonality. This video is a great explanation of orthogonal matrices, orthogonal bases, and Gram-Schmidt. A must see if you want a sort of wrap-up of orthogonal vectors, bases, matrices, projections, and other orthogonal topics.
