Page 205 - A Course in Linear Algebra with Applications

6.3: Kernel, Image and Isomorphism 189

L(V), the set of all linear operators on a vector space V over
a field F, is an algebra over F.
Suppose now that we pick and fix an ordered basis B for
the finite-dimensional vector space V. Then, with respect to
B, a linear operator T on V is represented by an n x n matrix,
which will be denoted by

M(T).

By 6.2.3 the matrix M(T) has the property

= M(r)[v] B .
[T(v)] B

It follows from 6.2.3 that the assignment of the matrix
M(T) to a linear operator T determines a bijective function
from L(V) to M n(F). The essential properties of this function
are summarized in the next result.
Theorem 6.3.10
Let Ti and T2 be linear operators on an n-dimensional vector
space V and let M(Ti) denote the matrix representing Ti with
respect to a fixed ordered basis B of V. Then the following
equations hold:
(i) M(Ti + T 2) = M(Ti) + M(T 2);
(ii) M{cT)=cM(T);
(iii) M(TiT 2 ) = M(Ti)M(T 2 ) for all scalars c.

It is as well to restate this technical result in words to
make sure that the reader grasps what is being asserted. Ac-
cording to part (i) of the theorem, if we add linear operators
T\ and T 2, the resulting linear operator T\ + T 2 is represented
by a matrix which is the sum of the matrices that represent
T\ and T 2. Also (ii) asserts that the scalar multiple cTi is
represented by a matrix which is just c times the matrix rep-
resenting T\.