Nonlinear Control Sy.. - Free

2.3. VECTOR SPACES 35
where the 1 element is in the ith row and there are zeros in the other n - 1 rows. Then
{el, e2i , en} is called the set of unit vectors in R. This set is linearly independent, since
Thus,
is equivalent to
Alel+...+Anen
Al
An
Ale, +...+Anen=0
A1=A2=...=An=0
Definition 2.4 A basis in a vector space X, is a set of linearly independent vectors B such
that every vector in X is a linear combination of elements in 13.
Example 2.4 Consider the space lR'. The set of unit vectors ei, i = 1, , n forms a basis
for this space since they are linearly independent, and moreover, any vector x E R can be
obtain as linear combination of the e, values, since
Ale, + ... + Anen =
It is an important property of any finite dimensional vector space with basis {bl, b2, , bn}
that the linear combination of the basis vectors that produces a vector x is unique. To prove
that this is the case, assume that we have two different linear combinations producing the
same x. Thus, we must have that
But then, by subtraction we have that
n n
x = E labz =
=1 z=1
n
i=1
r7:)bz=0
Al
A.n