Page 118 - A Course in Linear Algebra with Applications

102 Chapter Four: Introduction to Vector Spaces

for some scalars c$. If, on the other hand, no finite subset
generates V, then V is said to be infinitely generated.

Example 4.2.5
Show that the Euclidean space R n is finitely generated.
Let X\, X2, • •., X n be the columns of the identity matrix
In- If
a±
f \

A= ^
\a J
n
n
is any vector in R , then A = a\X\ + 0,2X2 + • • • + a nX n;
therefore X±,X2,.. • ,X n generate R n and consequently this
vector space is finitely generated.

On the other hand, one does not have to look far to find
infinitely generated vector spaces.

Example 4.2.6
Show that the vector space P(R) of all real polynomials in x
is infinitely generated.
To prove this we adopt the method of proof by contradic-
tion. Assume that P(R) is finitely generated, say by polyno-
mials Pi,P2, • • • iPki a n d look for a contradiction. Clearly we
may assume that all of these polynomials are non-zero; let m
be the largest of their degrees. Then the degree of any linear
combination of Pi,p2 • • • ,Pk certainly cannot exceed m. But
this means that x m+1 , for example, is not such a linear com-
bination. Consequently Pi,P2:---iPk do not generate P(R),
and we have reached a contradiction. This establishes the
truth of the claim.