Y= N/D D≠0
Hence if
denominator is zero then real number can never be possible.
So 1/0, 2/0, 3/0,………..are practically
impossible.
i.e. 1/0=
undefined.
Archimedean Property of real numbers
Theorem:- Let a be any real number and b any +ve real number. Then there exists a
positive integer n such that
nb>a
Proof:- Given that aϵ R & bϵ R+
So there are two possible cases.
Case 1:-
When a≤0
In
this case the relation nb>a is always true because the value of nb is always
+ve.
Case 2:- When a>0
Let
us that there exists no +ve integar such that nb>a
Then
we have nb≤a ⩝ n∈N
It means
that a be the upper bound of set S which is given by
S= {b,2b,3b,………..} = {nb:n∈N}.
Boundness of subsets of R
Upper bound of a subset
of R:-
Let S be a subset of real numbers. If there exist a real
number K , such that x≤K ⩝ x∈S
Then K is called an upper bound of set S.
If there
exists an upper bound for a set S then it is called “bounded above”.
Example:-
The set S= {………-4,-3,-2,-1} is bounded above & 9 is an upper bound.
Addition & Multiplication of Modulo - Abstract Algebra
Addition Modulo
Now we are going to discuss a new type of addition which is known as “addition modulo m” and written in the form
where a and b belongs to an integer and m is any fixed positive integer.By definition we have
Properties of Group
- The identity element of a group is always unique.
- The inverse of each element of a group is unique, i.e., in a group G with operation * for every
, there is only element 
such that
, e being the identity.
Table for Group- Abstract Algebra
The composition tables are useful in examining the following axioms in the manner explained below:
- Closure Axiom : If all the elements of the table belong to the set G (say) then G is closed under the Composition a (say). If any of the elements of the table does not belong to the set, the set is not closed.
Composition Table - Abstract Algebra
A Binary Operation in a finite set can completely be described with the help of a table. This table is well known as composition table. The composition table helps us to verify most of the properties satisfied by the binary operations. This table can be formed as follows:
Order of a Group- Abstract Algebra
Finite and infinite Groups:
If a group contains a finite number of distinct elements, it is called finite group otherwise an infinite group.
In other words, a group
is said to be finite or infinite according as the underlying set G is finite or infinite.
In other words, a group
is said to be finite or infinite according as the underlying set G is finite or infinite. Commutative Group or Abelian Group
If the commutative law holds in a group, then such a group is called an Abelian group or Commutative group. Thus the group
is said to be an Abelian group or commutative group if
, 
.
A group which is not Abelian is called a non-Abelian group. The group 
is called the group under addition while the group 
is known as group under multiplication.
is called the group under addition while the group is known as group under multiplication.
Algebraic Structure- Abstract Algebra
A non-empty set 
together with at least one binary operation defined on it is called an algebraic structure. Thus if G is a non-empty set and “
” is a binary operation on
, then 
is an algebraic structure.
are all algebraic structures. Since addition and multiplication are both binary operations on the set R of real numbers,
is an algebraic structure equipped with two operations.
together with at least one binary operation defined on it is called an algebraic structure. Thus if G is a non-empty set and “” is a binary operation on, then is an algebraic structure.are all algebraic structures. Since addition and multiplication are both binary operations on the set R of real numbers,
is an algebraic structure equipped with two operations.Identity and Inverse
Identity: A composition 
in a set
is said to possesses of an identity if there exists an element 
such that
Moreover, the element e, if it exists is called an identity element and the algebraic structure
is said to have an identity element with respect to
.
in a set is said to possesses of an identity if there exists an element such thatMoreover, the element e, if it exists is called an identity element and the algebraic structure
is said to have an identity element with respect to. Binary Operations
The concept of binary operation on a set is a generalization of the standard operations like addition and multiplication on the set of numbers. For instance we know that the operation of addition (+) gives for ally two natural numbers m,n another natural number m+n, similarly the multiplication operation gives for the pair m,n the number m,n in N again. These types of operations arc found to exist in many other sets. Thus we give the following definition.
Archimedean Property of Real Number
Let be any real number. Then there exists a natural number such that .
This theorem is known as the Archimedean property of real numbers. It is also sometimes called the axiom of Archimedes, although this name is doubly deceptive: it is neither an axiom (it is rather a consequence of the least upper bound property) nor attributed to Archimedes (in fact, Archimedes credits it to Eudoxus).
Vector Space (Theorem-5)
Statement:- The necessary & sufficient conditions for a non-empty subset W of a
vector spaceV(F) to be a subspace of V are
(
1) α∊W, β∊W ⇨ α-β∊W
(2) a∊F, α∊W ⇨ aα∊W
Proof:- Necessary Condition:-
Let V be
a vector space over the field F and W be its subspace.
∴
W will be a vector space over the same field F.
Field & Integral Domain
Integral domain:-
1-
A commutative ring with unity element and without zero divisors is called an
integral domain.
2-
An algebraic system (D,+,.) where D is a non-empty set with two binary
compositions to be denoted by addition and multiplication is called an integral
domain if following axioms are satisfied-
Definition of Ring
Ring:-
An algebraic system (R,+,.) consisting of a
non-empty set R with two binary composition (to be denoted by addition and
multiplication) is called a ring if following axiom are satisfied-
(R,+) is
an abelian group.
1- Closure axiom
2- Associative law
3- Identity element
4- Inverse axiom
(R,.) is
a semi group.
1- Closure axiom
2- Associative law
3- Multiplication distributive over
addition i.e.
a.(b+c)=a.b+a.c, ∀ a, b, c∊R
and (b+c).a=b.a+c.a, ∀a, b, c∊R
Linear Transformation Theorem-1
Statement:-
Let T be
a linear transformation from avector space U into V over the field F.Then T is
non-singular iff T is one-one.
Proof:-
Let T be
a non-singular transformation from U into V.
Let α1,α2∊U such that
Vector Space Question-1
Question:- Express (1,2,3) as a linear combination of (1,1,1),(2,-1,1) and (1,-2,5)
in V3(R).
Solution:- Let a1,a2,a3∊R such that
(1,2,3)=a1(1,1,1)+a2(2,-1,1)+a3(1,-2,5)……………………(a)
(1,2,3)=(a1,a1,a1)+(2a2,-a2,a2)+(a3,-2a3,5a3)
Definition of Group
An
algebraic system (G,o) where G be a non-empty set with o as defined binary
operation is called a group if following axiom are satisfied-
1- Closure Axiom:-
If a, b∊G ⇨ aob∊G , ∀ a,b∊G
then G
is said to be closed under the binary operation.
Binary Operation & Algebric Structure
Binary operation or Binary composition on a set:-
Let G be
a non-empty set then an operation ‘o’ on the non-empty set G is called binary
operation.
If a∊G, b∊G
⇒ aob∊G, ∀ a, b∊G
This
property is called closure property and if this is satisfied then G is said to
be closed under the binary composition ‘o’.
Real & Quadratic Forms
Bilinear Form:-
Let U and
V be two vector spaces over the same field F. A bilinear form on W=U⊕V is a fraction f from W into F,
which assigns to each element (α,β) in W a scalar f(α,β) in such a way that
f(aα1+bα2,β)=
af(α1,β)+bf(α2,β)
&
f(α,aβ1+bβ2)=af(α,β1)+bf(α,β2)
Here f(α,β) is an element of F. It denotes the
image of (α,β) under the function f. Thus a bilinear form on W is a function
from W into F which is linear as a function of either of its arguments when the
other is fixed.
Linear Transformation (Question-1)
Question:-
Show that the mapping T:V2(R)→V3(R)
defined by T(a,b)=(a+b, a-b, b) is a linear transformation from V2(R)
into V3(R).
Find the range, rank, null space and
null(T) of T.
Solution:-
Given that
T:V2(R)→V3(R)
Such that
T(a,b)=(a+b,a-b,b)
Vector Space (Theorem-4)
Question:-
Show that a field K can be regarded as a
vector space over any subfield F of K.
Solution:- Let K be a field and F be its subfield.
Since, K
is a field,
Therefore,
Union of two subspaces (Theorem)
Statement:-
The union
of two subspace of a vector space is a subspace iff one each contained.
Proof:-
Let V be
a vector space over the field F.Let W1 and W2 be its subspaces
such that either
Null Space, Rank, Nullity & Range
Range of a Linear Transformation:-
Let U and V
be two vector spaces over the field F.Let T be a linear transformation from U
into V then the of all vectors of V which are images of elements of U is called
range of linear transformation. It is denoted by R(T).
Thus R(T)={T(α)∊V:α∊U}
Annihilator Theorem-1
Theorem:-
Let V be a finite dimensional vector space over the field F and let W be a
subspace over the field F and let W be a subspace of V.Then W00=W.
Proof:-
We have
W0={f∊V’
:f(α)=0 ∀ α∊W}………..(1)
And W00={α∊V : f(α)=0 ∀ f∊W0}……..(2)
Definition of annihilator
Annihilators:-
If V is a
vector space over the field F and S is a subset of V,the annihilator of S is
the set So of all linear functional f on V such that
f(α)=0 ∀ α∊S
Sometimes A(S) is also used to denote the
annihilators of S.
Thus S0={f∊V’: f(α)=0 ∀ α∊S}
Annihilator Question-1
Question:-Let W1 and W2 be subspaces of a
finite dimensional vector space V.
(a)Prove that (W1+W2)o=W10⋂W20
(b)Prove that (W1⋂W2)0=W10+W20.
Solution:-
(a) First we shall prove that
W1⋂W2⊆(W1+W2)0.
One-One, onto & Invertible linear transformation
One-one transformation:-
Let T be
a transformation from a vector space U into V then T is said to be one-one
transformation if
α1,α2∊U
and α1≠α2 ⇒ T(α1)≠T(α2)
In other
word
α1,α2∊U
and T(α1)=T(α2) ⇒
α1=α2
Linear Transformation
Let U &
V be two vector spaces over the same field F.A linear transformation from U
into V is a function T:U→V, such that,
T(aα+bβ)=aT(α)+bT(β),for every a,b∊F and α,β∊U
This
condition is also called linearity property.
Theorem on linear sum
Statement:-
The linear
sum of two subspaces of a vector space is also a subspace of same vector space.
Linear Sum of two subspaces
Let V be a
vector space over the field F the linear sum of two subspaces W1 and
W2 of V written as (W1+W2) and is defined as W1+W2={α1+α2:α1єw1,α2єw2} which shows that each element 0f (W1+W2)
is expressible as sum of an element of W1 and an element of W2.
Linear Combination & Linear Span
Linear Combination Of Vectors:-
Let V be a
vector space over the field F,then a vector αєV is said to be a linear
combination of vectors α1,α2,α3,………………..αnєV.
If α=a1α1+a2α2+a3α3+…………………………+anαn
where
a1,a2,a3,……………………………….anєF
Invarience Theorem
Statement:- Any two bases
of a finite dimensional vector space have same number of elements.
OR
The number
of elements in a basis of a finite dimensional vector space is unique.
Intersection of two subspaces
Statement:-
The
intersection of any two sub-spaces of a vector space is also a subspace of the
same vector space.
Proof:-
Let V be
a vector space over the field F and W1 and W2 be its subspaces.
Inner Product Space
Let V be the
vector space over the field F where F is either field of real numbers or field
of complex number. An inner product space on V is a function from VχV into F which assigns to each
ordered pairs of vectors α,β in V by a scalar (α,β) such that
Existence Theorem
Statement:-There exists a basis for each finite
dimensional vector space.
Proof:-
Let V
be vector space & let S={ α1,α2,……………..,αm } be a finite subset of V such that L(S)=V. If S is
linearly independent then it is a basis of V.
Vector Space (Theorem-3)
Statement:- The
necessary and sufficient condition for a vector space V over the field F to be
direct sum of its two subspaces W1 and W2 are that
1-V=W1+W2
2- W1
and W2 are disjoint i.e. W1∩W2={0}
Direct Sum of a vector space
Let V be a vector space over the field F then
vector space V is said to be direct sum of its subspaces W1 and W2
written as V=W1⊕W2. If each element of V
is uniquely expressible as sum of an element of W1 and an element of W2.
In this case
W1 and W2 are called complementary subspaces.This definition can be extended for
more than two subspaces.
i.e. vector space V is said to be direct sum
of its subspaces W1,W2,W3,………………………………….,Wn if every element αєV can be written in one
and only one way
α=α1+α2+α3+…………………………..+αn
where α1єW1,α2єW2,α3єW3,…………………….αnєWn
Disjoint
subspaces:-
Two
subspaces W1 and W2 of a vector space V over the field F are said to be disjoint if their
intersection(∩) with zero subspace.
i.e. W1 andW2 are disjoint if W1∩W2={0}
Basis, Dimension & Cosets of a vector Space
Basis of
a vector space:-
A
non-empty subset S of a vector space V(F) is said to be its basis if
1-S is
linearly independent.
2-S
generates V i.e. L(S)=V
i.e. each
vector in V is expressible as a linear combination of element of S.
Vector Space (Theorem-2)
Theorem:- If W is a subspace of
an n-dimensional vector space over the field F then
dimW≤dimV
Proof:-
Let W be a subspace of a finite
dimensional vector space V(F).Let S={α1,α2,…………………..αm}
be a basis of V.
Veactor Space (Theorem-1)
Statement-The
necessary and sufficient condition for a non-empty subset W of vector space V
is that
a,bєF, α,βєW ⇨ aα+bβєW, ∀ a,bєF and α,βєW
Proof- Necessary condition-
Let W be
a subspace of a vector space V over the field F.
∵ W is a vector space over the same field F. So W is a closed under scalar multiplication.
Definition of Vector Space
Let F be an orbitrary field then a
non-empty set V is called a
vector space
over the field F written as V(F) if following axioms are satisfied-
❶-There is defined an internal composition
in V to be denoted additively such that (V,+) is an abelian group.
i.e.
1-Closure axiom:-
If α,β єV
then
α+βєV,
∀ α,βєBV
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