Page 201 - A Course in Linear Algebra with Applications

6.3: Kernel, Image and Isomorphism 185

The last result provides an alternative proof of the dimen-
sion formula in 6.3.3. Let T : V —> W be a linear transfor-
mation. Then dim(V/Ker(:T)) = dim(Im(T) by 6.3.7. Prom
the formula for the dimension of a quotent space (see 5.3.7),
we obtain
dim(F) - dim(Ker(T)) = dim(Im(T)),
so that dim(Ker(T)) + dim(Im(T)) = dim(V).

There is second isomorphism theorem which provides
valuable insight into the relation between the sum of two sub-
spaces and certain associated quotient spaces.
Theorem 6.3.8
If U and W are subspaces of a vector space V, then
(u + w)/w ~ u/(unw).

Proof
We begin by defining a function T : U —> (U + W)/W by
the rule T(u) = u + W, where u G U. It is a simple matter
to check that T is a linear transformation. Since u + W is a
typical vector in (U + W)/W, we see that T is surjective.
Next we need to compute the kernel of T. Now T(u) =
u + W equals the zero vector of (U + W)/W, i.e., the coset W,
precisely when u G W, which is just to say that u G U D W.
Therefore Ker(T) = UnW. It now follows directly from 6.3.7
that U/(U n l f ) ~ ( f / + W)/W.
We illustrate the usefulness of this last result by using it
to give another proof of the dimension formula of 5.3.2.
Corollary 6.3.9
// U and W are subspaces of a finite dimensional vector space
V, then
dim(C/ + W) + dim(U n W) = dim(C7) + dim(W).