Page 139 - A Course in Linear Algebra with Applications

5.1: Existence of a Basis 123

in a basis of V. But the latter must have n elements by 5.1.6,
and so it coincides with the set of Vj's.
Now assume that the vectors vi, v 2 ,..., v n generate V.
If these vectors are linearly dependent, then one of them, say
Vj, can be expressed as a linear combination of the others.
But this means that we can dispense with Vj completely and
generate V using only the v^'s for j ^ i, of which there are
n—1. Therefore dim(V) < n—1 by 5.1.2. By this contradiction
vi, V2,..., v n are linearly independent, so they form a basis
of V.

Example 5.1.7
The vectors

( - : ) • ( : ) • ( ! )
are linearly independent since the matrix which they form has
3
three pivots; therefore these vectors constitute a basis of R .
We conclude with an application of the ideas of this sec-
tion to accounting systems.

Example 5.1.8 (Transactions on an accounting system)
Consider an accounting system with n accounts, say
cti, CK2,..., oi n- At any instant each account has a balance
which can be a credit (positive), a debit (negative), or zero.
Since the accounting system must at all times be in balance,
the sum of the balances of all the accounts will always be zero.
Now suppose that a transaction is applied to the system. By
this we mean that there is a flow of funds between accounts of
the system. If as a result of the transaction the balance of ac-
count Q.i changes by an amount £$, then the transaction can be
represented by an n-column vector with entries t\,t2, • • •, t n.
Since the accounting system must still be in balance after the
transaction has been applied, the sum of the ti will be zero.