This document discusses isomorphism, one-to-one and onto linear transformations, and provides examples. It defines isomorphism as when two vector spaces are said to be isomorphic if there exists a linear transformation between them that is both one-to-one and onto. It provides theorems relating the dimension of isomorphic spaces and characterizations of one-to-one and onto linear transformations in terms of the kernel, rank, and nullity. Examples are given to determine whether linear transformations are one-to-one and/or onto based on their rank and nullity. Finally, several vector spaces are stated to be isomorphic to each other.

1. Shroff S.R. Rotary Institute of Chemical
Technology
PREPARED BY :-
1. RAJ PAREKH 140990119029
2. DWIJ PATEL 140990119032
3. KARAN PATEL 140990119033
SUBJECT: VECTOR CALCULUS AND LINEAR ALGEBRA
TOPIC : ISOMORPHISM

2.  Isomorphism:
other.eachtoisomorphicbetosaidare
andthen,tofrommisomorphisanexiststheresuch that
spacesvectorareandifMoreover,m.isomorphisancalledis
ontoandonetooneisthat:nnsformatiolinear traA
WVWV
WV
WVT 
 Thm 6.9: (Isomorphic spaces and dimension)
Pf:
.dimensionhaswhere,toisomorphicisthatAssume nVWV
onto.andonetooneisthat:L.T.aexistsThere WVT 
one-to-oneisT
nnTKerTT
TKer


0))(dim()ofdomaindim()ofrangedim(
0))(dim(
Two finite-dimensional vector space V and W are isomorphic if and only if
they are of the same dimension.

3. .dimensionhavebothandthatAssume nWV
onto.isT
nWT  )dim()ofrangedim(
nWV  )dim()dim(Thus
 
  .ofbasisabe,,,let
andV,ofbasisabe,,,Let
21
21
Wwww
vvv
n
n


nnvcvcvc
V
 2211
asdrepresentebecaninvectorarbitraryanThen
v
nnwcwcwcT
WVT


2211)(
follows.as:L.T.adefinecanyouand
v
It can be shown that this L.T. is both 1-1 and onto.
Thus V and W are isomorphic.

4. vector.singleaofconsistsrangein theevery w
ofpreimagetheifone-to-onecalledis:functionA WVT 
 One-to-one:
.thatimplies
)()(inV,vanduallforiffone-to-oneis
vu
vu

 TTT
one-to-one not one-to-one

5. Some important theorems related to one to
one transformation
 Thm 1: A linear transformation T : V -> W is one to one if and only if
ker(T) ={0}.
 Thm 2: A linear transformation T : V -> W is one to one if and only if
dim(ker(T)) = 0, i.e., nullity (T) = 0.
 Thm 3: A linear transformation T : V -> W is one to one if and only if
rank(T)=dim V.
 Thm 4: If A is an m x n matrix and TA : Rn -> Rn is multiplication by A then
TA is one to one if and only if rank (A)= n.
 Thm 5: If A is an n x n matrix and TA : Rn -> Rn is multiplication by A then
TA is one to one if and only if A is an invertible matrix.

6. inpreimageahasin
elementeveryifontobetosaidis:functionA
V
WVT
w

 Onto:
(T is onto W when W is equal to the range of T.)
Thm 1: A linear transformation T : V -> W is onto if and only if rank (T)
= dim W
Thm 2: If A is an m x n matrix and TA : Rn -> Rm is multiplication by A
then TA is onto if and only if rank (A) = m.
 Let T : V -> W be a linear transformation and let dim V = dim W
(i) If T is one-to-one ,then it is onto.
(ii) If T is onto, then it is one-to-one.

7.  Example :
neither.oronto,one,-to-oneiswhetherdetermineandof
rankandnullitytheFind,)(bygivenis:L.T.The
TT
ATRRT mn
xx 
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



100
110
021
)( Aa




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



00
10
21
)( Ab




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
110
021
)( Ac



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
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
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
000
110
021
)( Ad
Sol:
T:Rn→Rm dim(domain of T) rank(T) nullity(T) 1-1 onto
(a)T:R3→R3 3 3 0 Yes Yes
(b)T:R2→R3 2 2 0 Yes No
(c)T:R3→R2 3 2 1 No Yes
(d)T:R3→R3 3 2 1 No No

8.  (Isomorphic vector spaces)
space-4)( 4
Ra
matrices14allofspace)( 14 Mb
matrices22allofspace)( 22 Mc
lessor3degreeofspolynomialallofspace)()( 3 xPd
)ofsubspace}(numberrealais),0,,,,{()( 5
4321 RxxxxxVe i
The following vector spaces are isomorphic to each other.

9. Thank You