Page 195 - A Course in Linear Algebra with Applications

6.3: Kernel, Image and Isomorphism 179

cvi belong to Ker(T); thus Ker(T) is a subspace. For similar
reasons Im(T) is a subspace.

Let us look next at some examples which relate these new
concepts to some more familiar ones.

Example 6.3.1
Consider the homogeneous linear differential equation for a
function y of the real variable x:

y™ + a n_ x{x)y^-^ + ••• + a 1(x)y' + a 0(x)y = 0,

with x in the interval [a, b] and di(x) in .Doo[a, &]. There is
an associated linear operator T on the vector space D^a^b]
defined by

1
T(f) = / ( n ) + an-xOr)/*"- * + • • • + a x{x)f + a Q(x)f.

Then Ker(T) is the solution space of the differential equation.

Example 6.3.2
Let A be an m x n matrix over a field F. We have seen that
the rule T(X) = AX defines a linear transformation

Identify Ker(T) and Im(T).
In the first place, the definition shows that Ker(T) is
the null space of the matrix A. Next an arbitrary element of
Im(T) is a linear combination of the images of the standard
n
basis elements of R ; but the latter are simply the columns
of the matrix A. Consequently, the image of T coincides with
the column space of the matrix A.

Example 6.3.3
After the last example it is natural to enquire if there is an
interpretation of the row space of a matrix A as an image