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178 Chapter Six: Linear Transformations

6.3 Kernel, Image and Isomorphism
If T : V —> W is a linear transformation between two vec-
tor spaces, there are two important subspaces associated with
T, the image and the kernel. The first of these has already
been defined; the image of T,

Im(T),

is the set of all images T(v) of vectors v in V: thus Im(T) is
a subset of W.
On the other hand, the kernel of T

Ker(T)

is defined to be the set of all vectors v i n 7 such that T(v) =
0 W. Thus Ker(T) is a subset of V. Notice that by 6.2.1 the
zero vector of V must belong to Ker(T), while the zero vector
of W belongs to Im(T).
The first thing to observe is that we are actually dealing
with subspaces here, not just subsets.
Theorem 6.3.1
If T is a linear transformation from a vector space V to a
vector space W, then Ker(T) is a subspace of V and Im(T) is
a subspace of W.

Proof
We need to check that Ker(T) and Im(T) contain the relevant
zero vector, and that they are closed with respect to addition
and scalar multiplication. The first point is settled by the
equation T(Oy) = Ow, which was proved in 6.2.1. Also, by
definition of a linear transformation, we have T(vi + V2) =
T(vi) + T(v 2 ) and T(cvi) = cT(vi) for all vectors v 1 ; v 2 of
V and scalars c. Therefore, if vi and v 2 belong to Ker(T),
then T(vi + v 2 ) = 0 ^ , and T(cvi) = Ow, so that v x + v 2 and