Orthogonal Sets (12/2/05) Recall that “orthogonal” matches the geometric idea of “perpendicular”. Definition. A set of vectors u 1,u 2,…,u p in R n is called an orthogonal set if each pair of distinct vectors is orthogonal, i.e., u i  u j = 0 for all i  j. An orthogonal basis for a subspace W of R n is … (guess!)

Orthogonal bases are nice! Orthogonal bases are especially easy to work with since the weights of any vector with respect to that basis can be computed easily (no row reduction). Theorem. If u 1,u 2,…,u p is an orthogonal basis for W and y = c 1 u 1 + c 2 u 2 +…+ c p u p, then for each i, c i = y  u i / u i  u i. Example: Verify that (1,1) and (1,-1) form an orthogonal basis for R 2 and express (5,-2) in terms of this basis.

Orthogonal Projections Looking at the picture of the above example, we see that the weights we get give the orthogonal projection of the given vector onto the lines determined by the basis vectors. That is, if L is the line determined by a basis vector u, then the orthogonal projection of a vector y onto L is just proj L y = c u = (y  u / u  u) u.

Orthonormal Sets An orthogonal set all of whose vectors are unit vectors (recall, this means they have norm, or length, equal to 1) is called an orthonormal set. Note that orthonormal bases are even simpler than orthogonal bases since now each weight on y is just y  u. The standard basis e 1,e 2,…,e n for R n is obviously an orthonormal basis.

Orthogonal Matrices A square matrix is called orthogonal if its columns consist of orthonormal vectors. It’s easy to check that if U is an orthogonal matrix, then U T U = I n. Hence if U is orthogonal, its transpose and its inverse are the same!

Assignments Correct your test (due Wed 12/7) On Monday (12/5), we will have Lab #4 on diagonalization of matrices. Please read Section 5.3 in preparation for that lab. For Wednesday (12/7), please Read Section 6.2. Do Exercises 1-23 odd.