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Lecture 3

Introduction to Vector Space Theory

Matrices

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Linear Block Codes

matrixGeneratorG

(vector)word messagem(vector)word code

,

=

c

where

Gmc

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Vector Space-Introduction

An n -dimensional vector has a form

x = ( x1 , x2 , x3 , , x n ) . The set R n of n -dimensional vectors is a vector

space .

Any set V is called a vector space if it containsobjects that behave like vectors:

ie, they add & multiply by scalars according tocertain rules. In particular, they must be closed under vector addition and scalar multiplication .

But addition & scalar multiplication need not bedefined conventionally!

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Contd

Let V denote the vector space.The addition on Vis vector addition.The scalar multiplicationcombines a scalar from a Field F and a vectorfrom V. Hence V is defined over a field F.

V must form a commutative group under addition For any element a in F and any element v in V,a .V is an element in V.

Distributive law- a.(u+v)=a.u+a.v Associative law- (a.b).v=a.(b.v)

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Contd.

Important vector spaces:

R, R 2 , R 3, R n with usual + and scalar multn. M mn ; the set of all m x n matrices

Pn; all polynomials of degree n Consider a vector space over binary fieldF2.Consider the sequence u=u 0u n-1 where the

u i s are from {0,1}.We can construct such 2n

n-tuples over F2.Let Vn denote this set. Vn is aVector space over F2

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Subspaces

A set W of vectors is a subspace of vector space V if and only if W is a subset of V andW is itself a vector space under the sameaddition and scalar multiplication.

For any two vectors u,v W, (u+v) W .

For any element a in F and any u in W , a.umust be in W .

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Contd

To test if W is a Subspace

We should, but need not, check all the propertiesof a vector space in W : most hold because Wsvectors are also in the bigger vector space V .

But we must check closure in W : linear combinations of vectors in W must also lie in W .

This means the zero & additive inverses mustbe in W too.

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Examples

Let u 1,.,u k be a set of k vectors in V over

a field F. The set of all linear combinationsof u 1,.,u k forms a subspace of V. The set of polys of degree 2 or less is a

subspace of the set of polynomials of degree3 or less.

The set of integers is not a subspace of R,because the set of scalars includes fractions,eg 1/2.

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Spanning Sets &Linear Independence

A set S = { u 1,u 2,.......,u n } of vectors is said to span avector space V if every vector in V can beexpressed as a linear combination of the vectors inS.

Ex: ( x, y, z ) = x i + y j + z k , so every vector in R 3

isa linear combination of i, j & k . If any vector in a set can be expressed as a

linear combination of the others , we call theset linearly dependent . If not, the set is linearlyindependent .

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Basis set

A set of linearly independent vectors is a

basis for a Vector space V if each vector inV

can be expressed in one and only one way as alinear combination of the set.

In any Vector space or subspace there exists atleast one set B of linearly independent vectorswhich span the space.

The no. of vectors in the Basis of a Vectorspace is the dimension of the Vector space. One example of a basis are the vectors

(1,0,,0), (0,1,,0),, (0,0, , 1).

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Orthogonality

Let u= and

v=be two n-tuples in Vn. We define the inner product(dot product) as

u.v= where the multiplication and addition are

carried out in mod-2.. The inner product is a scalar. If u.v=0, then u and vare said to be orthogonal to each other

The inner product has the following properties

(1) u.v=v.u

(2) u.(v+w)=u.v+u.W

(3)(au).v=a(u.v).

),.....,( 110 nuuu

),....,( 110 nvvv

0 0 1 1 1 1........ n nu v u v u v + + +

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MatricesA k x n matrix over F2 is a rectangular array withk rows and n columns.

00 01 02 0, 1

10 11 12 1, 1

1,0 1,1 1,2 1, 1

.....

.....

. . . . .

. . . . .

.....

n

n

k k k k n

g g g gg g g g

G

g g g g

=

where each ijg 0 0i k and j n with

is an element from the binary field F2.

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G is also represented by its k rows

as0 0 1, ,..... k g g g

0

1

1

.

.k

g

g

G

g

=

Each row of G is an n-tuple and each column is a k-tuple over F2.

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If k (with ) rows of G are linearlyindependent , then the 2k linear combinationsof of these rows form a k dimensionalsubspace of the vector space Vn of all the n-

tuples over F2. This subspace is called therow space of G

Elementary row operations will not changethe row space of G

k n

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Let S be the row space of a k x n matrix G overF2 whose rows are linearly independent . Let Sd bethe null space of S. Then the dimension of Sd isn-k . Consider (n-k) linearly independent vectorsin Sd. These vectors span Sd. We can form an

(n-k) x n matrix H as00 01 02 0, 10

10 11 12 1, 11

1,0 1,1 1,2 1, 11

.....

.....

. . . . ..

. . . . .......

n

n

n k n k n k n k nn k

h h h hh

h h h hh

H

h h h hg

= =

The row space of H is Sd

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Since each row g i is a vector in S and each

row h j of H is a vector in Sd , the innerproduct of g i and h j must be zero. As therow space S of G is the null space of therow space Sd of H, S is called the null spaceor dual space of H.

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Theorem For any k x n matrix G over F2, with k linearly

independent rows, there exists an (n-k) x nmatrix over the same field with (n-k) linearlyindependent rows such that for any row g i inG and any h j in H, gi.hj = 0 . The row space ofG is the null space of H and vice versa.