MAT 2401Linear Algebra

5.3 Orthonormal Bases

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HW

WebAssign 5.3 Written Homework

Basis

S={(1,0,0),(0,1,0),(0,0,1)} is the standard basis for R3 . It is described as an orthonormal basis.

Every element in R3 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 …

JPEG is not possible without …

Basis

Not all basis are created equal!

Good, Better, Best

1 2

1 2

1 2

, , , Basis

, , , Othogonal Basis

, , , Othonormal Basis

n

n

n

v v v

w w w

u u u

Orthonormal Bases

A basis S for an inner product space V is orthonormal if1. For u,vS, <u,v>=0.2. For uS, u is a unit vector.

Example 1

S={(1,0),(0,1)} is an orthonormal basis for R2 with the dot product.(From previous lecture, we know S is a basis of R2)

Example 1

S={(1,0),(0,1)} is an orthonormal basis for R2 with the dot product.

Remark

The standard basis is an orthonormal basis for Rn with the dot product.

Example 2

S={1, x, x2} is an orthonormal basis for P2 with the usual inner product.

Example 2 S={1, x, x2}

Example 3

S={(1,1,1), (-1,1,0), (1,2,1)} is a basis for R3. However, it is not orthonormal.

Q: How to “get” a orthonormal basis from S?

Gram-Schmidt Process

1 2

1 2

1 2

, , , Basis

, , , Othogonal Basis

, , , Othonormal Basis

n

n

n

v v v

w w w

u u u

Gram-Schmidt Process

Idea

Example 3

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

Example 3

Example 3