Chapter 1.2 Definitions (Vector Spaces)

All Definitions in Chapter 1.2

Vector Space

A ______ _____, over a field F, consists of a set on which 2 operations (addition and scalar multiplication) are defined so that for each pair of elements x,y in V, there is a unique element x+y in V, and for each element a in F and each element x in V, there is a unique element ax in V, such that (VS 1)…(VS 8) hold.

VS 1

For all x,y∈V, x+y=y+x (commutativity of addition)

VS 2

For all x,y,z∈V, (x+y)+z=x+(y+z) (associativity of addition)

VS 3

There exists an element in V denoted by 0 such that x+0=x for each x∈V.

VS 4

For each element x∈V, there exists an element y∈V such that x+y=0

VS 5

For each element x∈V, 1x=x

VS 6

For each pair of elements a,b∈F and each element x∈V, (ab)x=a(bx)

VS 7

For each element a∈F and each pair of elements x,y∈V, a(x+y)=ax+ay

VS 8

For each pair of elements a,b∈F and each element x∈V, (a+b)x=ax+bx

Scalars

Elements of the field F

Vectors

Elements of the vector space V

n-tuple

An object of the form (a1, a2, …, an) where the entries a1, a2, …,an are elements of a field F. Also, the elements a1, a2, …,an are called the entries/components of the _-____

Equivalent n-tuples

Two _-______ (a1, a2, …, an) and (b1, b2, …, bn) with entries from a field F are called equal if ai=bi for i=1,2,…,n

Column Vector

Has the form |a1|

|a2|

….

|an|

Row Vector

Has the form (a1, a2, …, an)

Matrix

An mxn ______ with entries from a field F is a rectangular array of the form:

|a11 a12 … a1n|

|a21 a22 … a2n|

|…………………|

|am1 am2 … amn|

Diagonal Entries

Entries have the form aij. ________ _______ of a matrix are entries where i=j.

i=row number and j=column number

Zero Matrix

The mxn matrix where all entries are 0

Square Matrix

An mxn matrix where m=n

Equivalent Matrices

Two mxn matrices A and B are called equal if all their entries are the same.

Matrix Addition and Scalar Multiplication

(A+B)ij = Aij + Bij, and (cA)ij = cAij

Equivalent Functions

Two functions f,g∈F(S,F) are called equal if f(s)=g(s) for each s in S.

Polynomial

A __________ with coefficients from a field F is an expression of the form

f(x) = an*x^n + an-1*x^n-1 + … +a1x + a0

where n is a non-negative integer and each ak, called the coefficient of xk, is in F.

Zero Polynomial

A polynomial where f(x)=0, where

an = an-1= a0 = 0

and its degree is defined to be -1

Degree

The ______ of a polynomial is the largest exponent of x with a nonzero coefficient.

Equivalent Polynomials

Two polynomials

f(x) = an*x^n + an-1*x^n-1 + … + a1x + a0

g(x) = bm*x^m + bm-1*x^n-1 + … + b1x + b0

are called equal if m=n and ai=bi for i=0,1,…,n

Sequence

A ________ in F is a function sigma from the positive integers n into F, meaning

Input: n=1,2,3,…

Output: An element in F, which can be a complex or real number

Theorem 1.1: Cancellation Law

If x,y,z are vectors in a vector space V such that x+z=y+z, then x=y.

Theorem 1.2

In any vector space V, the following statements are true:

a) 0x=0 for each x in V

b) (-a)x=-(ax)=a(-x) for each a in F and each x in V

c) a0=0 for each a in F