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MAC 2103

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Module 11

lnner Product Spaces II

Upon completing this module, you should be able to:

- Construct an orthonormal set of vectors from an orthogonal set of vectors.
- Find the coordinate vector with respect to a given orthonormal basis.
- Construct an orthogonal basis from a nonstandard basis in ℜⁿ using the Gram-Schmidt process.
- Find the least squares solution to a linear system Ax = b.
- Find the orthogonal projection on col(A).
- Obtain the best approximation.

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Rev.F09

The major topics in this module:

Orthogonal Bases, Gram-Schmidt Process,

Least Squares and Best Approximation

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Rev.09

We have learned from the previous module that two vectors u and v in an inner product space V are orthogonal to each other iff <u,v> = 0.

To obtain an orthonormal set, we will normalize each of the vectors in the orthogonal set.

How to normalize the vectors? This can be done by dividing each of them by their respective norm and making each of them a unit vector.

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Rev.F09

Example 1: Find the orthonormal set of vectors from the following set of vectors: Let S ={v1, v2} where v1 = (5,0) and v2= (0,-3).

Step 1: Verify that the set of vectors are mutually orthogonal with respect to the Euclidean inner product on ℜ².

Step 2: Find the norm for both vectors.

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Rev.F09

Step 3: Normalize the vectors in the orthogonal set.

Step 4: Verify that the set S is orthonormal by showing that

and

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Rev.F09

Orthonormal Set: An orthogonal set in which each vector is a unit vector.

Orthonormal basis: A basis consisting of orthonormal vectors in an inner product space. Example: The standard basis for ℜⁿ.

Orthogonal basis: A basis consisting of orthogonal vectors in an inner product space.

Note that if S is an orthogonal set, then S is a linearly independent set.

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Rev.F09

If S ={v1, v2 , … , vn} is an orthogonal basis of W, then for any w∈ W,

where

are called the Fourier coefficients.

So the coordinate vector of w,

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Rev.F09

If S ={q1, q2 , … , qn} is an orthonormal basis of W, then for any w∈ W,

where are called the Fourier coefficients.

So the coordinate vector of w,

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Rev.F09

Example 2: Compute the coefficients and determine the coordinate vectors in Example 1 for u = (10,3).

From Example 1, we have v1 = (5,0), v2= (0,-3) and

In this case, the coefficients are:

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Rev.F09

So the coordinate vector of u,

We can see that a nice advantage of working with an orthogonal basis is that the coefficients in any basis representation for a vector are immediately known; they are called Fourier coefficients.

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Rev.F09

Example 3: Find the coordinates of w = (2,3) relative to the orthonormal basis for ℜ², B = {v1, v2}, where

Since B is orthonormal, we have

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Rev.F09

Let S = {u1, u2, …, um} with nonzero ui∈ℜⁿ for i = 1, 2, … , m. S does not have to be a linearly independent set. It might be that A = [u1u2 … um] is an n x m matrix and the source of S.

The Gram-Schmidt Algorithm: S = {u1, u2, …, um}

Let

For k = 2, 3, … , m, let

If we discard it since it is linearly dependent.

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Rev.F09

We then have r ≤ m orthogonal and linearly independent vectors in B = {v1, v2, …, vr}.

span(S) = span(B) = W, a r dimensional subspace of ℜⁿ and B is an orthogonal basis for W. W = col(A) if A = [u1u2 … um].

An orthogonal basis for W is {q1, q2, …, qr} where

for i =1, 2, … , r.

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Rev.F09

Let ℜⁿ be the usual Euclidean inner product space of dimension n. Let {u1, u2, …, un} be a nonstandard basis in ℜⁿ.

Step 1: Use the Gram-Schmidt method to construct an orthogonal basis {v1, v2, …, vn} from the basis vectors {u1, u2, …, un}.

Step 2: Normalize the orthogonal basis vectors to obtain the orthonormal basis {q1, q2,…, qn}.

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Rev.F09

Example 4:

The set is a nonstandard basis in ℜ³.

Step 1:

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Rev.F09

form an orthogonal basis for ℜ³.

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Rev.F09

Step 2: Normalize the orthogonal basis vectors to obtain the orthonormal basis B = {q1, q2, q3}.

The norms of these vectors are:

So an orthonormal basis is B where

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Rev.F09

The linear system Ax = b always has the associated normal system ATAx = ATb which is consistent and has one or more solutions. Any solution of this system is a least squares solution of Ax = b. Moreover, the orthogonal projection of b on W = col(A) is where x is a least squares solution.

Example 5: Find the least squares solution of the linear system Ax = b given by

Observe that A has two linearly independent column vectors, so ATA is invertible, and there is a unique solution to ATAx = ATb, which will be our least squares solution to Ax = b.

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Rev.F09

We have

So the normal system ATAx = ATb in this case is

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Rev.F09

By Gauss Elimination, the row-echelon form of

is

The solution to the normal system ATAx = ATb in this case is

is our unique least square solution to Ax = b.

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Rev.F09

is the orthogonal projection of b on W = col(A). The is the Best Approximation to b in W since the distance between b and is the minimum for all vectors in W,

Example 6: From the previous example, the orthogonal projection of b on W = col(A) is

Thus, we have obtained the best approximation to b in W.

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Rev.F09

We have learned to :

- Construct an orthonormal set of vectors from an orthogonal set of vectors.
- Find the coordinate vector with respect to a given orthonormal basis.
- Construct an orthogonal basis from a nonstandard basis in ℜⁿ using the Gram-Schmidt process.
- Find the least squares solution to a linear system Ax = b.
- Find the orthogonal projection on col(A).
- Obtain the best approximation.

http://faculty.valenciacc.edu/ashaw/

Click link to download other modules.

Rev.F09

Some of these slides have been adapted/modified in part/whole from the following textbook:

- Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition

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Rev.F09