LESSON 11: Orthogonal Sets and the Gram-Schmidt Process

Terminology:

inner (or dot) product, length (or norm) of a vector, unit vector, normalize, distance between vectors, orthogonal vectors, orthogonal complement, angle between vectors, orthogonal set, orthogonal basis, orthonormal basis, orthogonal projection, Gram-Schmidt process.

Objectives:

learn about length of vectors and distance between vectors; learn about orthogonal vectors; learn how to determine an orthonormal basis for a set of vectors.

Reading Assignment:

Chapter 6, Sections 6.1-6.2, 6.4 (pages 295-386, 397-399 only).

Lesson Outline

Key Ideas and Discussion:

The formulas for length and distance in are straightforward generalizations of the formulas for that are usually learned in high school. A vector is normalized by dividing by the vector by its length, so that the result is a unit vector. Taking the dot product of two vectors provides an easy test for orthogonality (the generalization to higher dimensions of the concept of perpendicularity).

If you have a set of mutually orthogonal vectors (an orthogonal set) taken from some vector space, then the set is always a basis (an orthogonal basis) for some subspace of the vector space. An orthonormal basis is an orthogonal basis with the additional property that all of the basis vectors are normalized. The standard basis for is an orthonormal basis that you should all be very familiar with. An orthonormal basis for a vector space is very easy to work with, because only dot products are needed to determine the coordinates for any vector in the space, relative to the basis. Given any basis for a subspace, the Gram-Schmidt provides an organized method for finding an orthonormal basis.

Practice Problems:

6.1.1, 5, 7, 11, 13, 17, 19 (pp. 376-377);
6.2.3, 9, 13, 17, 23 (pp. 386-387); 6.4.3, 5, 7, 11 (pp. 402-403).

Assignment