Page 141 - A Course in Linear Algebra with Applications

5.1: Existence of a Basis 125

one unit from account a.\ to account an and which does not
affect other accounts.

Exercises 5.1
3
1. Show that the following sets of vectors form bases of R ,
and then express the vectors Ei, E2, E3 of the standard basis
in terms of these:

(b) Y t = 1 , Y 2 = 1 , Y 3 =

2. Find a basis for the null space of each of the following
matrices:

1 - 5 2 3 1 1
(a) | - 4 2 - 6 J ; (b) 3 1 4 - 7
3 1 1 2 1 0

3. What is the dimension of the vector space M mjTl (F) where
F is an arbitrary field of scalars?

4. Let V be a vector space containing vectors vi, V2,..., v n
and suppose that each vector of V has a unique expression
as a linear combination of i, V2,..., v n . Prove that the Vj's
v
form a basis of V.
5. If S is a subspace of a finitely generated vector space V,
establish the inequality dim(S') < dim(V).