MAT 2401 Linear Algebra 5.3 Orthonormal Bases

HW WebAssign 5.3 Written Homework

Basis S={(1,0,0),(0,1,0),(0,0,1)} is the standard basis for R 3. It is described as an orthonormal basis. Every element in R 3 can be written as a linear combination of elements in S. (3,4,-2)=3i+4j-2k In general, we can consider this process as encoding “a piece of info” by the elements in the basis.

Preview Orthonormal basis is fundamental to the development of Fourier Analysis and Wavelets which have all kind of applications such as signal processing, image compression, and processing. We will look at how to find orthonormal bases for a inner product space V.

Quote… It is very difficult to show you why, in practical applications, we want this specific kind of bases. So I am going to show you an excerpt from chapter 6 of the book “The World According to Wavelets” by Barbara Hubbard.

Quote…about Efficiency The fact that all the vectors in a non- orthogonal basis come into play for the computation of a single coefficient is also bothersome when one wants to compute or adjust quantization errors.

Quote…about Efficiency In an orthogonal basis one can calculate the “energy” of the total error by adding the energies of the errors of each coefficient; it’s not necessary to reconstruct the signal. In a non-orthogonal basis one has to reconstruct the signal to measure the error.

Quote…about Redundancy The most dramatic comparison is between … “everything is said 10 times.” In an orthonormal basis, each vector encodes information that is encoded nowhere else.

Another Example…jpeg

JPEG is not possible without …

Basis Not all basis are created equal!

Good, Better, Best

Orthonormal Bases A basis S for an inner product space V is orthonormal if 1. For u,v  S, =0. 2. For u  S, u is a unit vector.

Example 1 S={(1,0),(0,1)} is an orthonormal basis for R 2 with the dot product. (From previous lecture, we know S is a basis of R 2 )

Example 1 S={(1,0),(0,1)} is an orthonormal basis for R 2 with the dot product.

Remark The standard basis is an orthonormal basis for R n with the dot product.

Example 2 S={1, x, x 2 } is an orthonormal basis for P 2 with the usual inner product.

Example 2 S={1, x, x 2 }

Example 3 S={(1,1,1), (-1,1,0), (1,2,1)} is a basis for R 3. However, it is not orthonormal. Q: How to “get” a orthonormal basis from S?

Gram-Schmidt Process

Idea

Example 3 S={(1,1,1), (-1,1,0), (1,2,1)}

Example 3