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4.1 Spaces and Subspaces 161

less, it is important to recognize some of the more significant examples and to
understand why they are indeed vector spaces.
Example 4.1.1
Because (A1)–(A5) are generalized versions of the five additive properties of
matrix addition, and (M1)–(M5) are generalizations of the five scalar multipli-
cation properties given in §3.2, we can say that the following hold.
m×n
• The set of m × n real matrices is a vector space over .
m×n
• The set C of m × n complex matrices is a vector space over C.
Example 4.1.2
The real coordinate spaces




x 1
 
 
 
1×n n×1  x 2 
= {( x 1 x 2 ··· x n ) ,x i ∈ } and =  .  ,x i ∈
.
 . 
 
 
 
x n
are special cases of the preceding example, and these will be the object of most
of our attention. In the context of vector spaces, it usually makes no difference
whether a coordinate vector is depicted as a row or as a column. When the row or
column distinction is irrelevant, or when it is clear from the context, we will use
n
the common symbol to designate a coordinate space. In those cases where
it is important to distinguish between rows and columns, we will explicitly write
1×n n×1
or . Similar remarks hold for complex coordinate spaces.
Although the coordinate spaces will be our primary concern, be aware that
there are many other types of mathematical structures that are vector spaces—
this was the reason for making an abstract definition at the outset. Listed below
are a few examples.
Example 4.1.3
With function addition and scalar multiplication defined by

(f + g)(x)= f(x)+ g(x) and (αf)(x)= αf(x),

the following sets are vector spaces over :
• The set of functions mapping the interval [0, 1] into .
• The set of all real-valued continuous functions defined on [0, 1].
• The set of real-valued functions that are differentiable on [0, 1].
• The set of all polynomials with real coefficients.