Vector Spaces

Goals: To understand the definition of a Linear Space and to be able to determine if a given set is a linear space.

I. The Game of Mathematics

The Two Basic Rules of the Game: 1) Begin with some definitions and axioms as a foundation, then using the laws of logic, build a mathematical system . 2) The system cannot be illogical. As we have seen with fields, the definitions and axioms of a system oftentimes are properties extracted from an already familiar system. At this point it may be helpful to think of Linear Spaces as an abstraction of some of the properties of vectors in the plane R².

From this point on Linear Spaces, also called Vector Spaces, will be our principle object of study.

II. Vectors in R²

A nonzero vector x in the plane can be visualized geometrically by an arrow.

Once a coordinate system is defined on the plane a nonzero vector whose tail is at the origin may be represented by the coordinates of the point at it's head.

That is,

Every vector in the plane that has the same direction and magnitude (length) as x is equal to x. Consequently, for our purposes, we only need to consider vectors whose tail is located at the origin.

The magnitude or length of a vector may be calculated from it's coordinates by application of the Pythagorean theorem.

Scalar Multiplication

Given the vector x in the plane, we illustrate scalar multiplication as follows:

In terms of the coordinate representation, given:

we have

and

,

,

or in general,

if

and

is any real number, then

.

The scalar multiplication ax then acts like a stretching or shrinking operation on a vector x, and reverses the direction of the arrow if a < 0. Scalar multiplication is one of the properties of vectors in the plane that we will use in the definition of a Linear Space.

Vector Addition and Subtraction

The next property that we need for our definition of Linear Spaces is that of addition and subtraction. Addition and subtraction of vectors in the plane is visualized as:

If we are given the coordinate representaions for the vectors u and v, then vector addition is defined as:

Given,

and

then

.

Vector subtraction is defined as:

Given,

and

then

.

From the illustration above, if

then

or

With these properties in mind we are ready to define Linear Spaces.

III. The Definition of a Vector Space

Perhaps the simplest of all mathematical systems is one in which two elements may be "added" together to produce another element. If we think of scalar multiplication as combining the same element with itself, we have the basic ingredients for a Linear Space.

Definition: A Vector Sapce or Linear Space V over a field F is a set whose elements we call vectors, that have an addition property and a scalar multiplication property that satisfy the following eight axioms:

Given, u, v, w in V, and a, b in F

A1: u + v = v + u for any u, v in V.
A2: (u + v) + w = u + (v + w) for any u, v, w in V.
A3: There exists an element O in V such that u + O = u, for each u in V.
A4: For each u in V, there exists a -u in V such that u + (-u) = O.
A5: a(u + v) = a u + av for each scalar and any u, v in V.
A6: (a + b)u = au + bu for any scalars a, b and any u in V.
A7: (ab)u = b(bu) for any scalars a and b, and any u in V.
A8: 1*u = u for all u in V.

Some of you may recognize that the first four of the vector space axioms can be summarized by saying that V is an abelian group under addition. Also, if you have studied vector calculus, you may be familiar we the Euclidean Vector spaces R² , R³ and Rⁿ. If so, you can think of "Abstract Vector Spaces" as sets that satisfy the same vector space axioms as Euclidean vector spaces. Notice, however, that Linear Spaces do not require that the vectors have length.
The term Linear Space is an older term (circa 1920) for what today is generally referred to as a Vector Space. The terms are synonymous.

III. Examples of Vector Spaces

Although there are many sets that qualify as vector spaces, we will primarily focus on four types. We will use these four types to demonstrate the power of the concepts and techniques that we will study over the remainder of the course.

Akamai: When determining if a given set is a vector space, be able to identify: the Set whose elements are called vectors, the Field , how vector addition and scalar multiplication are defined, which element of the set is the zero vector and that the given set satisfies the eight vector space axioms.

1) The Vector Space M^nxm

You might be surprised at first to learn that the matrices of equal dimension form a vector space.

The Set: All M^nxm matrices.

The Field: Generally one of the following fields:The real numbers, The Rational Numbers, The Complex Numbers or a finite field Z_p.

A Vector: A single M^nxm matrix.

Vector Addition: Matrix Addition.

Scalar Multiplication: Scalar multiplication

The zero vector: The zero matrix.

Show that the set of M^nxm Matrices forms a vector space.

An Example of n x 1 Matrices over Z₂

An important example of a vector space we will see in computer science, is the n x 1 matrices, or column vectors over Z₂, that is, vectors whose entries are either 0 or 1. We will see that such a vector space is used for error correcting codes, image compression and the efficient encoding and decoding of information.

If n = 7, then vectors look like:

and

.

Check the axioms for this vector space.

2) The Vector Space P_n

The Set: All single variable polynomial functions of degree less than n.

The Field: The real numbers

A Vector: A single polynomial of degree less than n.

Vector Addition: Addition of polynomials.

Scalar Multiplication: Scalar multiplication

The zero vector: The zero polynomial.

Show that the set P₃, forms a vector space.

Let

and

3) The Vector Space C[a,b]

We will examine this vector space in the context of signal processing, with Fourier analysis and wavelets. This vector space is extremely important in Digital Signal Processing, think just about everything from radio and TV, DVD, cell phones, communication and the internet to Cat scans, Seti and prediction of the tides.

The Set: The set of all continuous functions over the closed interval [a,b].

The Field: The real numbers

A Vector: A single continuous function.

Vector Addition: Addition of functions.

Scalar Multiplication: Scalar multiplication

The zero vector: The zero function.

Show that the set C[a,b], forms a vector space.

Let

and

be any two vectors in C[a,b].

Define vector addition as:

4) The Vector Space Rⁿ

These vector spaces are also known as Euclidean vector spaces. R² is used to model the plane, while R³ is used to model three dimensional space. The calculus of several variables is created within this vector space.

The Set: The set of all nx1 column matrices for a given n. (in our case)

The Field: The real numbers R

A Vector: A column vector.

Vector Addition: Addition of matrices.

Scalar Multiplication: Matrix scalar multiplication

The zero vector: The nx1 zero matrix.

5) The Vector Space Z._pⁿ

These vector spaces are composed of n x 1 row vectors over the finite field Z_p. For example, you are probably familiar with term byte from computer science. A byte is a string composed of eight bits (binary digits) of either 0 or 1. The set of all 256 possible bytes along with the properties from the finite field Z₂ form the vector space denoted as Z₂⁸.

The Set: The set of all 1xn row matrices for a given n. (in our case)

The Field: Numbers from the finite field Z_p.

A Vector: A row vector.

Vector Addition: Addition of matrices.

Scalar Multiplication: Matrix scalar multiplication

The zero vector: The 1xn zero matrix.

Show Z₂⁸ is a vector space.

Vectors look like:

and

Vector addition:

Scalar multiplication:

or

Show Z₅⁴ is a vector space.

Exercise.

IV. Additional Properties

Theorem: If V is a vector Space and x is any element of V, then

i) 0 x = 0

ii) x + y = 0 implies y = -x

iii) (-1)x = -x

Proof:

In class.

____________________________________________________

Lecture 6 Notes