09. General Vector Spaces - Yonsei Universityweb.yonsei.ac.kr/hgjung/Lectures/MAT203/09 General Vector... · 2014-12-29 · 9.1. Vector Space Axioms Vector Space Axioms The 10 properties

E-mail: [email protected]://web.yonsei.ac.kr/hgjung

9. General Vector Spaces9. General Vector Spaces


9.1. Vector Space Axioms9.1. Vector Space Axioms

Vector Space Axioms

The 10 properties in this definition are called the vector space axioms, and a set V with two operations that satisfy these 10 axioms is called a vector space.



Function Spaces

If u=(u1, u2, …, un) is a vector in Rn, then we can regard the components of u to be the values of a function f whose independent variable varies over the integers from 1 to n; that is,

To continue the analogy between n-tuples and functions, let us denote the set of real-valued functions that are defined for all real values of x by F(-∞, ∞).f=g if and only if f(x)=g(x) for all real values of x

(cf)(x)=cf(x)

(f+g)(x)=f(x)+g(x)



Subspaces



Subspaces

Let n be a nonnegative integer, and let Pn be the set of all real-valued functions of the formExample 1Example 1

where a0, a1, …, an are real numbers; that is, Pn is the set of all polynomials of degree n or less. Show that Pn is a subspace of F(-∞, ∞).

Polynomials are defined for all real values of x, so Pn is a subset of F(-∞, ∞).

Let and

Thus,

and

which shows that cp and p+q are polynomials of degree n or less.



Subspaces

Cm(-∞, ∞): the real-valued functions with continuous mth derivatives



Subspaces

The subspace of F(-∞, ∞) that is spanned by the functions 1, x, x2, …, xn consists of all linear combinations of these functions and hence consists of all functions of the form

Example 11, 12Example 11, 12

These are the polynomials of degree n or less, so span{1, x, x2, …, xn}=Pn

These functions are linearly independent and hence form a basis for Pn.

This is called the standard basis for Pn.



Wroński’s Test for linear Independence of Functions

Suppose that f1(x), f2(x), …, fn(x) are functions in C(n-1)(-∞, ∞).

If these functions are linearly dependent, then there exist scalars k1, k2, …, kn that are not all zero and such that

for all x in the interval (-∞, ∞).




which is called the Wronskian of f1(x), f2(x), …, fn(x), is zero for every x in the interval (-∞, ∞).

This implies that the linear system

has a nontrivial solution for every x in the interval (-∞, ∞).

And this, in turn, implies that the determinant




This function is nonzero for all real values of x and hence is not identically zero on (-∞, ∞). Thus, the functions are linearly independent.

Example 14Example 14f1(x)=1, f2(x)=ex, and f3(x)=e2x



Dimension

We saw in Example 12 that the function 1, x, x2, …, xn form a basis for Pn. This implies that all bases for Pn have n+1 vectors and hence that dim(Pn)=n+1.

Also, we showed above that P∞ is infinite-dimensional, so it follows that F(-∞,∞), C(-∞,∞), C1(-∞,∞), Cm(-∞,∞), and C∞(-∞,∞) are also infinite-dimensional.

Example 15Example 15


9.2. Inner Product Spaces; Fourier Series9.2. Inner Product Spaces; Fourier Series

Inner Product Axioms

To give a vector space a geometric structure we need to have a third operation that will generalize the concept of dot product.

length, distance, angle, and orthogonality




We make the following definition whose statements duplicate those in Theorem 1.2.6, but in different notation.






The Effect of Weighting on Geometry

If f and g are continuous functions on the interval [a, b], then the formulaExample 4 (The Integral Inner Product)Example 4 (The Integral Inner Product)

defines an inner product on the vector space C[a, b] that we will call the integral inner product.

The norm of a function f relative to this inner product is

(7)

(8)



Axiom 1 – If f and g are continuous functions on [a, b], thenExample 4 (The Integral Inner Product)Example 4 (The Integral Inner Product)

Axiom 2 – If f, g, and h are continuous functions on [a, b], then

Axiom 3 – If f and g are continuous functions on [a, b] and k is a scalar, then

Axiom 4 – If f is a continuous function on [a, b], then




If p and q are distinct positive integers, then the functions cospx and cosqx are orthogonal with respect to the inner product

Example 5Example 5




Algebraic Properties of Inner Products







REMARK

From (30) of Section 1.2 for the angle θ between nonzero vectors u and v can be extended to general inner product spaces are



Orthogonal Bases

Recall that Theorem 7.9.1 that an orthogonal set of nonzero vectors in Rn is linearly independent. The same is true in a general inner product space V.

Example 8Example 8

Trigonometric polynomial

If cn and dn are not both zero, the f(x) is said to have order n.

The set of all trigonometric polynomials of order n or less is the subspace of C(-∞,∞) that is spanned by the functions in the set

We will denote this subspace by Tn.

Show that S is an orthogonal basis for Tn with respect to the integral inner product.



Orthogonal Bases

The set S spans Tn.

And, S is an orthogonal set:

Example 8Example 8



Orthogonal Bases

Find an orthonormal basis for Tn relative to the inner product.Example 9Example 9

Thus, normalizing the vectors in S yields the orthonormal basis

(21)



Best Approximation



Fourier Series

Let f be a continuous function on the interval [0, 2π].

We can compute the best mean square approximation to f by a trigonometric polynomial of order n or less. We will use the orthonormal basis (21) in Example 9.

Let the orthogonal projection of f on Tn be

(26)



Fourier Series

Thus,



Fourier Series

or more briefly,

The numbers a0, a1, …, an, b1, …, bn are called the Fourier coefficients of f, and (26) is called the nth-order Fourier approximation of f.

It can be proved that if f is continuous on the interval [0, 2π], then the mean square error in the nth-order Fourier approximation of f approaches zero as n∞. This is denoted by writing

which is called the Fourier series for f.



General Inner Products on Rn

Axiom 1 – Since uTAv is a 1×1 matrix, it is symmetric and hence

Axiom 2 – Using properties of the transpose we obtain



General Inner Products on Rn

and

Axiom 3 – Again using properties of the transpose we obtain

Axiom 4 – Since A is positive definite and symmetric, the expression <v, v>=vTAv is a positive definite quadratic form, and hence Definition 8.4.2 implies that

if and only if

Weighted Euclidean inner product


9.3. General Linear Transformations; Isomorphism9.3. General Linear Transformations; Isomorphism

General Linear Transformations



Kernel and Range



Properties of The Kernel and Range



Isomorphism

We will show that Rn is “the mother of all real finite-dimensional vector spaces” in the sense that every real n-dimensional vector space differs from Rn only in the notation used to represent vectors.

The fact that the transformation T between a vector space V and n-tuples in Rn is one-to-one, onto, and linear means that it matches up V with Rn in such a way that operations on vectors in either space can be performed using their counterparts in the other space.



Thus, although a polynomial a0+a1x+a2x2 is obviously a different mathematical object from an ordered triple (a0, a1, a2), the vector spaces formed by these objects have the same algebraic structure.

Example 17Example 17The following table shows how the transformation

matches up vector operations in P2 and R3.

Isomorphism



The fact that every real n-dimensional vector space is isomorphic to Rn makes it possible to apply theorems about vectors in Rn to general finite-dimensional vector spaces.

Isomorphism



Consider the differentiation operatorExample 20Example 20

on the vector space of polynomials of degree 3 or less.

If we map P3 and P2 into R4 and R3, respectively, by the natural isomorphisms, then the transformation D produces a corresponding transformation

For example,

Isomorphism



Since Example 20Example 20Example 20Example 20

and hence the standard matrix for T is

Isomorphism



This matrix performs the differentiation Example 20Example 20Example 20Example 20

by operating on the images of the polynomials under the natural isomorphisms, as confirmed by the computation

Isomorphism



Inner Product Space Isomorphisms

If V and W are inner product spaces, then we call an isomorphism T:VW an inner product space isomorphisom if

It can be proved that if V is any real n-dimensional inner product space and Rn has the Euclidean inner product (the dot product), then there exists an inner product space isomorphism from V to Rn.

Under such an isomorphism, the inner product space V has the same algebraic and geometric structure as Rn.



Let Rn be the vector space of real n-tuples in comma-delimited form, let Mn be the vector space of real n×1 matrices,

let Rn have the Euclidean inner product <u, v>=u∙v,

and let Mn have the inner product <u, v>=uTv in which u and v are expressed in column form.

The mapping T:RnMn defined by

Example 21Example 21

is an inner product space isomorphism, so the distinction between the inner product space Rn

and the inner product space Mn is essentially notational, a fact that we have used many times in this text.

Inner Product Space Isomorphisms

Documents

09. General Vector Spaces - Yonsei Universityweb.yonsei.ac.kr/hgjung/Lectures/MAT203/09 General Vector... · 2014-12-29 · 9.1. Vector Space Axioms Vector Space Axioms The 10 properties