Nonlinear Control Sy.. - Free

2.3. VECTOR SPACES 33
(3) 3 0 E X : x + 0 = x ("0" is the neutral element in the operation of addition).
(4) 3 - x E X : x + (-x) = 0 (Every x E X has a negative -x E X such that their sum
is the neutral element defined in (3)).
(5) 31 E F : 1.x = x ("1 " is the neutral element in the operation of scalar multiplication).
(6) \(x + y) = .\x + Ay (first distributive property).
(7) (A + µ)x = Ax + µx (second distributive property).
(8) A (µx) = (Aµ) x (scalar multiplication is associative).
A vector space is called real or complex according to whether the field F is the real or complex
number system.
We will restrict our attention to real vector spaces, so from now on we assume that F =
R. According to this definition, a linear space is a structure formed by a set X furnished
with two operations: vector addition and scalar multiplication. The essential feature of the
definition is that the set X is closed under these two operations. This means that when
two vectors x, y E X are added, the resulting vector z = x + y is also an element of X.
Similarly, when a vector x E X is multiplied by a scalar a E IR, the resulting scaled vector
ax is also in X.
A simple and very useful example of a linear space is the n-dimensional "Euclidean"
space 1Rn consistent of n-tuples of vectors of the following form:
x
More precisely, if X = IRn and addition and scalar multiplication are defined as the usual
coordinatewise operation
xl
X2
xn
xl yl xl + yl xl Axl
X2 + y2 de f x2 + y2 Ax = A X2 de f Ax2
L xn Jn xn + yn xn Axn J
then it is straightforward to show that 1Rn satisfies properties (1)- (8) in Definition 2.2. In
the sequel, we will denote by xT the transpose of the vector x, that is, if
x
xl
X2