Nonlinear Control Sy.. - Free

34 CHAPTER 2. MATHEMATICAL PRELIMINARIES
then xT is the "row vector" xT = [xi, x2, , x,,]. The inner product of 2 vector in x, y E
R is xT y = E° 1 x: y:
Throughout the rest of the book we also encounter function spaces, namely spaces
where the vectors in X are functions of time. Our next example is perhaps the simplest
space of this kind.
Example 2.1 Let X be the space of continuous real functions x = x(t) over the closed
interval 0 < t < 1.
It is easy to see that this X is a (real) linear space. Notice that it is closed with respect
to addition since the sum of two continuous functions is once again continuous.
2.3.1 Linear Independence and Basis
We now look at the concept of vector space in more detail. The following definition
introduces the fundamental notion of linear independence.
Definition 2.3 A finite set {x2} of vectors is said to be linearly dependent if there exists a
corresponding set {a,} of scalars, not all zero, such that
On the other hand, if >2 atx, = 0 implies that a2 = 0 for each i, the set {x,} is said to be
linearly independent.
Example 2.2 Every set containing a linearly dependent subset is itself linearly dependent.
Example 2.3 Consider the space R" and let
0
et= 1 1
0