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.
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.