1 mac 2103 module 11 lnner product spaces ii. 2 rev.f09 learning objectives upon completing this...

1

MAC 2103Module 11

lnner Product Spaces II

2Rev.F09

Learning Objectives

Upon completing this module, you should be able to:

1. Construct an orthonormal set of vectors from an orthogonal set of vectors.

2. Find the coordinate vector with respect to a given orthonormal basis.

3. Construct an orthogonal basis from a nonstandard basis in ⁿℜ using the Gram-Schmidt process.

4. Find the least squares solution to a linear system Ax = b. 5. Find the orthogonal projection on col(A).6. Obtain the best approximation.

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3Rev.09

General Vector Spaces II


Orthogonal Bases, Gram-Schmidt Process,Orthogonal Bases, Gram-Schmidt Process,Least SquaresLeast Squares and and Best ApproximationBest Approximation

The major topics in this module:


4Rev.F09

How to Construct an Orthonormal Set of Vectorsfrom an Orthogonal Set of Vectors?


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.


5Rev.F09

How to Construct an Orthonormal Set of Vectorsfrom an Orthogonal Set of Vectors? (Cont.)


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 ²ℜ .

Step 2: Find the norm for both vectors.

⟨rv1,rv2⟩=

rv1 ⋅rv2 = (5)(0) + (0)(−3) = 0 S is an orthogonal set.

rv1 = 52 + 02 =5, andrv2 = 02 + (−3)2 =3.


6Rev.F09

How to Construct an Orthonormal Set of Vectorsfrom an Orthogonal Set of Vectors? (Cont.)


Step 3: Normalize the vectors in the orthogonal set.

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

rq1 =

rv1rv1=55,05

⎛⎝⎜

⎞⎠⎟=(1,0),

rq2 =rv2rv2=03,−33

⎛⎝⎜

⎞⎠⎟=(0,−1)

rq1 =

rq2 =1. ⟨rq1,rq2⟩= 0

⟨rq1,rq2⟩=

rq1 ⋅rq2 = (1)(0) + (0)(−1) = 0,

rq1 = ⟨rq1,rq1⟩

12 = (rq1 ⋅

rq1)12 = 12 + 02 = 1, and

rq2 = ⟨rq2 ,rq2⟩

12 = (rq2 ⋅

rq2 )12 = 02 + (−1)2 = 1.

7Rev.F09

Orthonormal Set, Orthonormal Basis, and Orthogonal Basis


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

ⁿ.ℜ

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.

8Rev.F09

Orthonormal Set, Orthonormal Basis, and Orthogonal Basis (Cont.)


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,

rw = ⟨

rw, rvi ⟩⟨rvi,rvi ⟩i=1

n

∑ rvi =⟨rw, rv1⟩⟨rv1,rv1⟩rv1 +

⟨rw, rv2 ⟩⟨rv2 ,rv2 ⟩rv2 + ...

⟨rw, rvn ⟩⟨rvn,rvn ⟩rvn,

rwS =(

rw)S =⟨rw, rv1⟩⟨rv1,rv1⟩,⟨rw, rv2 ⟩⟨rv2 ,rv2 ⟩, ... ,

⟨rw, rvn ⟩⟨rvn ,rvn ⟩

⎛⎝⎜

⎞⎠⎟.

⟨rw, rv1⟩

⟨rv1,rv1⟩,⟨rw, rv2⟩

⟨rv2 ,rv2⟩, ... ,

⟨rw, rvn⟩

⟨rvn ,rvn⟩

9Rev.F09

Orthonormal Set, Orthonormal Basis, and Orthogonal Basis (Cont.)


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,

rw = ⟨rw, rqi ⟩⟨rqi,rqi ⟩i=1

n

∑ rqi = ⟨rw, rqi ⟩

i=1

n

∑ rqi

=⟨rw, rq1⟩

rq1 + ⟨rw, rq2 ⟩

rq2 + ... ⟨rw, rqn ⟩

rqn,

as ⟨rqi,rqi ⟩ =

rqi2 =1 for i =1,2, ... ,n.

rwS =(

rw)S =(⟨rw, rq1⟩, ⟨

rw, rq2 ⟩, ... , ⟨rw, rqn ⟩).

⟨rw, rq1⟩,⟨

rw, rq2⟩, ... ,⟨rw, rqn⟩

10

Rev.F09

How to Find the Coordinate Vector with Respect to a Given Orthogonal Basis?


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:

⟨ru, rv1⟩

⟨rv1,rv1⟩=ru ⋅ rv1rv1

2 =(10)(5) + (3)(0)

52=5025= 2

⟨ru, rv2⟩

⟨rv2 ,rv2⟩=ru ⋅ rv2rv2

2 =(10)(0) + (3)(−3)

32=−99= −1

rv1 =5, and

rv2 =3.

11

Rev.F09

How to Find the Coordinate Vector with Respect to a Given Orthogonal Basis? (Cont.)


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.

ruS =(

ru)S =⟨ru, rv1⟩⟨rv1,rv1⟩,⟨ru, rv2 ⟩⟨rv2 ,rv2 ⟩

⎛⎝⎜

⎞⎠⎟=(2,−1).

12

Rev.F09

How to Find the Coordinate Vector with Respect to a Given Orthonormal Basis?


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

Since B is orthonormal, we have

rv1 =(−

12,−12), and rv2 =(

12,−12).

⟨rw, rv1⟩=

rw ⋅ rv1 = (2,3) ⋅(−12,−12) = (2)(−

12) + 3(−

12) = −

52,

⟨rw, rv2⟩=

rw ⋅ rv2 = (2,3) ⋅(12,−12) = (2)(

12) + 3(−

12) = −

12,

and rwB = (rw)B = (⟨

rw, rv1⟩,⟨rw, rv2⟩) = (−

52,−12).

13

Rev.F09

The Gram-Schmidt Process


Let S = {u1, u2, …, um} with nonzero ui ∈ ⁿ for i = 1, 2, … , ℜm. S does not have to be a linearly independent set. It might be that A = [u1 u2 … 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.

rvk =

ruk −⟨ruk,rvi ⟩

⟨rvi,rvi ⟩i=1

k−1

∑ rvi.

rv1 =

ru1.

rvk =

r0,

14

Rev.F09

The Gram-Schmidt Process (Cont.)


3. 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 ⁿ ℜand B is an orthogonal basis for W. W = col(A) if A = [u1 u2 … um].

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

for i =1, 2, … , r. rqi =

rvirvi

15

Rev.F09

How to Construct an Orthonormal Basis from a Nonstandard Basis in ⁿ ℜ using the Gram-Schmidt Process?


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

ⁿ.ℜ

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}.

16

Rev.F09

How to Construct an Orthonormal Basis from a Nonstandard Basis in ⁿ ℜ using the Gram-Schmidt Process? (Cont.)


Example 4: The set is a nonstandard basis

in ³.ℜ Step 1:

{ru1,ru2 ,ru3}={(−1,1,0),(1,2,1),(3,1,1)}

rv1 =ru1 =(−1,1,0),

rv12 =2, W1 =span({

rv1}), dim (W1)=1.

Projectionsonto W1 have one component.rv2 =

ru2 −projW1 (ru2 )=(1,2,1)−

⟨ru2 ,rv1⟩

rv12

rv1

=(1,2,1)−12(−1,1,0)=(

32,32,1),

rv22 =112, W2 =span({

rv1,rv2}) dim (W2 )=2.

Projectionsonto W2 have two components.

17

Rev.F09



form an orthogonal basis for ³. ℜ

rv3 =ru3 −projW2 (

ru3)=(2,1,1)−⟨ru3,rv1⟩

rv12

rv1 −⟨ru3,rv2 ⟩

rv22

rv2

=(3,1,1)−(−2)2(−1,1,0)−

711 / 2

(32,32,1)

=(3,1,1)+ (−1,1,0)−1411(32,32,1)

=(3,1,1)+ (−1,1,0)−(2111,2111,1411)=(

111,111,−311).

Thus, rv1 =(−1,1,0),rv2 =(

32,32,1),

rv3 =(111,111,−311).

18

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 rv1 = 2,

rv2 =112, and rv3 =

111.

rq1 =rv1rv1=12(−1,1,0)=(−

12,12,0),

rq2 =rv2rv2=

211(32,32,1)=(

3 22 11

,3 22 11

,211), and

rq3 =rv3rv3= 11(

111,111,−311)=(

1111,1111,−3 1111

).

19

Rev.F09

How to Find the Least Squares Solution?


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.

A =

1−21

0−14

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥and

rb =

−1−27

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

Arx =projW (

rb),

20

Rev.F09

How to Find the Least Squares Solution? (Cont.)


We have

So the normal system ATAx = ATb in this case is

ATA =1 −2 1

0 −1 4

⎡

⎣⎢⎢

⎤

⎦⎥⎥

1−21

0−14

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= 6 6

6 17⎡⎣⎢

⎤⎦⎥

ATrb =

1 −2 1

0 −1 4

⎡

⎣⎢⎢

⎤

⎦⎥⎥−1−27

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= 10

30⎡⎣⎢

⎤⎦⎥

6 66 17

⎡⎣⎢

⎤⎦⎥

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 10

30⎡⎣⎢

⎤⎦⎥

21

Rev.F09

How to Find the Least Squares Solution? (Cont.)


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.

ATA | AT

rb⎡⎣ ⎤⎦=

6 66 17

1030

⎡⎣⎢

⎤⎦⎥

x1 =−5 / 33, and x2 =20 /11.

1 10 1

−5 / 3320 /11

⎡⎣⎢

⎤⎦⎥.

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=rx =(x1, x2 )=(−5 / 33,20 /11)

22

Rev.F09

How to Find the Orthogonal Projection and Obtain the Best Approximation?


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. projW (

rb)=Arx =

1−21

0−14

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−5 / 3320 /11

⎡⎣⎢

⎤⎦⎥=

−5 / 33−50 / 33235 / 33

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

Arx =projW (

rb)

rb −projW (

rb) ≤

rb −rw for all rw ∈W .

projW (rb)

projW (rb)

23

Rev.F09

What have we learned?

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

ⁿℜ 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.



24

Rev.F09

CreditSome 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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