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8/6/2019 itc Lecture3
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Lecture 3
Introduction to Vector Space Theory
Matrices
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Block codes: basic definitions An alphabet is a discrete (usually finite) set of
symbols.example: B = { 0; 1} is the binary alphabet
Definition: A block code of blocklength n over an
alphabet X is a nonempty set of n-tuples ofsymbols from X.
The n-tuples of the code are called codewords.
Codewords are vectors whose components aresymbols in X.
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Block codes: basic definitions Codewords of length n are typically generated
by encoding messages of k information bitsusing an invertible encoding function.
Number of codewords is M = 2k , Rate R = k/n
The rate is a dimensionless fraction; the fractionof transmitted symbols that carry information.
A code with blocklength n and rate k/n is calledan (n; k) code
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Linear Block Codes
matrixGeneratorG
(vector)wordmessagem
(vector)wordcode
,
=
c
where
Gmc
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Vector Space-Introduction
Ann-dimensional vector has a form
x = (x1, x2, x3, , xn ). The setRn ofn-dimensional vectors is a vector
space.
Any set Vis called a vector space if it containsobjects that behave like vectors:
ie, they add & multiply by scalars according to
certain rules. In particular, they must beclosedunder vector addition and scalar multiplication.
Butaddition &scalar multiplication need not be
defined 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, R2, R3,Rn with usual + andscalar multn. Mmn; the set of allm xn matrices
Pn; all polynomials of degree n Consider a vector space over binary fieldF2.Consider the sequence u=u0un-1 where the
ui sare 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 Wof vectors isasubspace of vectorspace Vif and only ifW is a subset ofV andW is itself a vector space under the same
addition 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 ifWis a Subspace
We should, but need not, check all the propertiesof a vector space in W: most hold because Ws
vectors are also in the bigger vector space V.
But wemust check closure in W: linear
combinations of vectors in Wmust also lie in W.
This means thezero &additive inverses mustbe in Wtoo.
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Examples
Let u1,.,uk be a set of k vectors in V over
a field F. The set of all linear combinationsof u1,.,uk 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.
Theset of integers isnot a subspace ofR,because the set of scalars includes fractions,eg 1/2.
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Spanning Sets &Linear Independence
A set S = {u1,u2,.......,un} of vectors is said tospan avector space Vifevery vector in Vcan beexpressed as a linear combination of the vectors inS.
Ex:(x, y, z ) = x i + yj + z k, so every vector inR
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 dimensionof 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 v
are 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 with
k 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, ,..... kg 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 spaceVn 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 overF2whose rows are linearly independent. Let Sd be
the null space ofS. Then the dimension ofSd is
n-k. Consider (n-k) linearly independentvectorsin 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 gi is a vector in S and each
row hj of H is a vector in Sd, the innerproduct of gi and hj must be zero. As therow space S of G is the null space of the
row space Sd of H, S is called the null space
or dual spaceof 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 gi in
G and any hj in H, gi.hj = 0. The row space ofG is the null space of H and vice versa.