Linear Algebra, Theory And Applications, 2012a

8.3. LOTS OF FIELDS 205
such that
0 =
m∑
c i v i
i=1
and not all the c i are equal to zero. Suppose c k ≠0. Then the vector, v k may be solved for
in terms of the other vectors. Consequently,
V =span(v 1 , ··· , v k−1 , v k+1 , ··· , v m )
contradicting the definition of m. This proves the first part of the theorem.
To obtain the second part, begin with {u 1 , ··· , u k } and suppose a basis for V is
{v 1 , ··· , v n } .
If
span (u 1 , ··· , u k )=V,
then k = n. If not, there exists a vector,
u k+1 /∈ span (u 1 , ··· , u k ) .
Then by Lemma 8.2.10, {u 1 , ··· , u k , u k+1 } is also linearly independent. Continue adding
vectors in this way until n linearly independent vectors have been obtained. Then
span (u 1 , ··· , u n )=V
because if it did not do so, there would exist u n+1 as just described and {u 1 , ··· , u n+1 }
would be a linearly independent set of vectors having n+1 elements even though {v 1 , ··· , v n }
is a basis. This would contradict Theorem 8.2.4. Therefore, this list is a basis.
8.2.3 The Basis Of A Subspace
Every subspace of a finite dimensional vector space is a span of some vectors and in fact it
has a basis. This is the content of the next theorem.
Theorem 8.2.12 Let V be a nonzero subspace of a finite dimensional vector space, W of
dimension, n. ThenV has a basis with no more than n vectors.
Proof: Let v 1 ∈ V where v 1 ≠ 0. If span {v 1 } = V, stop. {v 1 } is a basis for V .
Otherwise, there exists v 2 ∈ V which is not in span {v 1 } . By Lemma 8.2.10 {v 1 , v 2 } is a
linearly independent set of vectors. If span {v 1 , v 2 } = V stop, {v 1 , v 2 } is a basis for V. If
span {v 1 , v 2 } ̸= V, then there exists v 3 /∈ span {v 1 , v 2 } and {v 1 , v 2 , v 3 } is a larger linearly
independent set of vectors. Continuing this way, the process must stop before n + 1 steps
because if not, it would be possible to obtain n + 1 linearly independent vectors contrary to
the exchange theorem, Theorem 8.2.4.
8.3 Lots Of Fields
8.3.1 Irreducible Polynomials
I mentioned earlier that most things hold for arbitrary fields. However, I have not bothered
to give any examples of other fields. This is the point of this section. It also turns out that
showing the algebraic numbers are a field can be understood using vector space concepts