Page 201 - Matrix Analysis & Applied Linear Algebra

196 Chapter 4 Vector Spaces

the columns of B are linearly independent, and hence N (B)= {0} —recall
(4.3.2). Therefore, the supposition that there exists a basis for V containing
fewer than n vectors must be false, and we may conclude that B is indeed a
minimal spanning set.
Proof of (4.4.2) =⇒ (4.4.1). If B is a minimal spanning set, then B must
be a linearly independent spanning set. Otherwise, some b i would be a linear
combination of the other b ’s, and the set

B = {b 1 ,..., b i−1 , b i+1 ,..., b n }
would still span V, but B would contain fewer vectors than B, which is im-

possible because B is a minimal spanning set.
Proof of (4.4.3) =⇒ (4.4.1). If B is a maximal linearly independent subset
of V, but not a basis for V, then there exists a vector v ∈V such that
v /∈ span (B) . This means that the extension set

B∪ {v} = {b 1 , b 2 ,..., b n , v}
is linearly independent—recall (4.3.15). But this is impossible because B is a
maximal linearly independent subset of V. Therefore, B is a basis for V.
Proof of (4.4.1) =⇒ (4.4.3). Suppose that B is a basis for V, but not a
maximal linearly independent subset of V, and let
Y = {y 1 , y 2 ,..., y k }⊆V, where k> n

be a maximal linearly independent subset—recall that (4.3.16) insures the ex-
istence of such a set. The previous argument shows that Y must be a basis
for V. But this is impossible because we already know that a basis must be a
minimal spanning set, and B is a spanning set containing fewer vectors than Y.
Therefore, B must be a maximal linearly independent subset of V.
Although a space V can have many different bases, the preceding result
guarantees that all bases for V contain the same number of vectors. If B 1 and
B 2 are each a basis for V, then each is a minimal spanning set, and thus they
must contain the same number of vectors. As we are about to see, this number
is quite important.

Dimension
The dimension of a vector space V is defined to be

dim V = number of vectors in any basis for V
= number of vectors in any minimal spanning set for V
= number of vectors in any maximal independent subset of V.