One-to-one (Injective): A Linear Transform T : V → W is called one-to-one if
the preimage of every w in the range consists of a single vector.
T is one-to-one iff u, v V , T (u ) T (v) u v
Theorem A: A Linear Transform T : V → W is one-to-one if and only if
Ker(T)={0}.
Proof: → If T is 1-1,
Let v Ker (T ) T (v) 0 , but for any L.T. always T(0) = 0,
as 0 = 0, so by definition of 1-1, T v T 0 v 0 Ker (T ) {0}.
← Let Ker(T)={0}, we have to show T is 1-1.
Let T (u ) T (v)
T (u ) T (v) 0
T (u v) 0 u v Ker (T )
But Ker(T)={0}, so u v 0 u v , so T is 1-1.
Onto (Surjective): A Linear Transform T : V → W is said to be onto if every
element in W has a preimage in V.
Note: T is onto, when W is equal to the range of T (ImT).
Theorem-B: A Linear Transform T : V → W is onto if and only if
Rank(T)=DimW.
Theorem-C: If T : V → W is Linear Transform with vector space V & W both of is
of same dimension. Then is T is 1-1 if and only if it is onto.
Invertible: A Linear Transform T : V W is said to be invertible T 1 : W V , if
inverse of the function is possible.
Theorem-D: A linear transformation T : V → W is invertible if and only if it is
injective (1-1) and surjective (onto).
Isomorphism: If a linear transformation T : V → W is invertible (1-1 & onto)
then the linear transformation T called Isomorphism and vector spaces V and
W are called isomorphic to each other.
We use V W symbol for isomorphic.
Theorem-E: Let V and W are two vector spaces V and W. If T : V → W is
invertible Linear Transform, then T-1 : W → V also linear.
Theorem-F: Two vector spaces V and W are if and only if Dim(V) = Dim(W).
Proof: Assume that V W , and Dim(V) = n.
So, by definition, there exists a L.T. T : V →W that is one to one and onto.
(1). As T is 1-1, then Ker(T)=0,
Then by Rank-Nullity Theorem for L.T. T, we have
Rank(T)+ Ker(T) = n
Rank(T)+ 0 = n
Rank(T) = n
(2). As T is onto, Rank(T) = Dim(W), then we get Dim(W)= n Dim(W) Dim(V) n
#
Proof of converse part: Assuming that Dim(W) Dim(V) n ,
Therefore, let v1, v2 , v3 ,....vn is basis of V, and w1 , w2 , w3 ,....wn is basis of W.
Then any arbitrary vector v V can be represented as; v 1v1 2v2 3v3 .... nvn
so there exists a linear transform T : V →W such that T (vi ) wi ; i 1,2,3...., n
Taking L.T. T on v 1v1 2v2 3v3 .... nvn , we get
T (v) 1T (v1 ) 2T (v2 ) 3T (v3 ) .... nT (vn ) 1w1 2 w2 3w3 .... n wn
T is 1-1??: Let v, v* V , so we can generate them by the basis of V as
v 1v1 2v2 3v3 .... nvn & v* 1*v1 2*v2 3*v3 .... n*vn .....(1)
Let T (v) T ( v* ) T (1v1 2v2 3v3 .... nvn ) T (1*v1 2*v2 3*v3 .... n*vn )
1T (v1 ) 2T (v2 ) 3T (v3 ) .... nT (vn ) 1*T (v1 ) 2*T (v2 ) 3*T (v3 ) .... n*T (vn )
1 1* T (v1 ) 2 2* T (v2 ) 3 3* T (v3 ) .... n n* T (vn ) 0
1 1* w1 2 2* w2 3 3* w3 .... n n* wn 0
w1, w2 , w3 ,....wn is basis of W and hence it is L.I. Therefore i i* 0; i 1,2,3...n
So, i i* by eqn(1), v v* SO the T is 1-1.
T is onto??: let w W
n
n n n
Then w i wi T ( w) T i wi T ( w) iT wi T ( w) ivi v
i 1 i 1 i 1 i 1
It means for for any w there exist v, such that T(v) = w. So T is onto.
Hence V W .
Isomorphic Vector Spaces examples:
HW
We can also find inverse L.T. of given L.T. by matrix as
Composition of two linear transforms:
Let T : U → V and S : V → W are two linear transformations, then there
composition is also form a linear transformation given by
SoT(x)=S[T(x)]
Note: SoT ToS
Question: Find SoT and ToS for the given transformations T : R3 → R2 and
S : R2 → R3 defined as
T ( x, y , z ) ( x y z , x z )
S ( x, y ) ( x, x y , y )
(The standard matrix of a composition)
