Page 108 - A Course in Linear Algebra with Applications

92 Chapter Four: Introduction to Vector Spaces

the vector space of all n-column vectors with entries in C.
More generally it is to carry out the same construction with
an arbitrary field of scalars F, in the sense of 1.3; this yields
the vector space

of all n-column vectors with entries in F, with the usual rules
of matrix addition and scalar multiplication.

Vector spaces of matrices
One obvious way to extend the previous examples is by
allowing matrices of arbitrary size. Let

M m?n (R)

denote the set of all m x n matrices with real entries. This
set is closed with respect to matrix addition and scalar mul-
tiplication, and it includes the zero matrix 0 m>n . The rules of
matrix algebra guarantee that M miTl (R) is a vector space. Of
m
course, if n = 1, we recover the Euclidean space R , while if
m = 1, we obtain the vector space

of all real n-row vectors. It is consistent with notation estab-
lished in 1.3 if we write

M n (R)

for the vector space of all real n x n matrices, instead of
M n (R). Once again R can be replaced by any field of scalars
n
F in these examples, to produce the vector spaces

M m , n (F), M n(F) and F n.