C Linear or Vector Spaces
143
Definition C.2 The inner product on a vector space V is an operation, which
associates with any pair x and y of elements of V a scalar x, y (real or complex) and
has the following properties: (1) x, x > 0 unless x = 0 in which case x, x = 0,
(2) x, y = =y, x ∗ , and (3) α x, y = =α x, y and x + z, y = =x, y + +z, y,
where z is a vector in V and the symbol ∗ denotes complex conjugate.
Definition C.3 The inner product in the Euclidian space R n is
x, y =
n
i=1
x i y i , x, y ∈ R
n ,
(C.6)
where {x i } and {y i } denote the components of x and y. If x, y = 0, the vectors x
and y in V are said to be orthogonal.
Example C.4 The operations with vectors of the physical space considered in
Example C.1 satisfy the properties of the inner product since (see Eq. C.6)
x, y = x 1 y 1 + x 2 y 2 + x 3 y 3 ,
(C.7)
and the vector addition and multiplication by scalars is performed component-bycomponent. Note that x, x = x 2
1 + x 2
2 + x 2
3 is the square of the length of x, which
is zero if and only if x = 0 is the null vector.
Example C.5 Similar arguments hold for the finite-dimensional vector space of
Example C.2 since x, y =
n
i=1 x i y i , and the vector addition and multiplication
by scalars is performed component-by-component. As mentioned previously, the
square of the length of x is
n
i=1 x 2
i so that x, x = 0 if and only if x = 0.
Example C.6 The values of the elements f and g of the vector space of functions
in Example C.3 at arbitrary t can be viewed as “components” of these infinitedimensional vectors. The extension of Eq. C.6 to the continuous case yields the inner
product
f, g =
τ
0
f (t) g(t) dt.
(C.8)
That this definition satisfies the properties of the inner product follows from the
operations defined by Eq. C.5. Note also that the square of the “length” of f is
f, f =
τ
0 f (t) 2 dt so that it is zero if only if f (t) = 0, t ∈ [0, τ ].
Definition C.4 A set of vectors x r , r = 1, . . . , p, of R n is linearly independent if
p
r=1
α r x r = 0 implies α r = 0, r = 1, . . . , p,
(C.9)
143
Definition C.2 The inner product on a vector space V is an operation, which
associates with any pair x and y of elements of V a scalar x, y (real or complex) and
has the following properties: (1) x, x > 0 unless x = 0 in which case x, x = 0,
(2) x, y = =y, x ∗ , and (3) α x, y = =α x, y and x + z, y = =x, y + +z, y,
where z is a vector in V and the symbol ∗ denotes complex conjugate.
Definition C.3 The inner product in the Euclidian space R n is
x, y =
n
i=1
x i y i , x, y ∈ R
n ,
(C.6)
where {x i } and {y i } denote the components of x and y. If x, y = 0, the vectors x
and y in V are said to be orthogonal.
Example C.4 The operations with vectors of the physical space considered in
Example C.1 satisfy the properties of the inner product since (see Eq. C.6)
x, y = x 1 y 1 + x 2 y 2 + x 3 y 3 ,
(C.7)
and the vector addition and multiplication by scalars is performed component-bycomponent. Note that x, x = x 2
1 + x 2
2 + x 2
3 is the square of the length of x, which
is zero if and only if x = 0 is the null vector.
Example C.5 Similar arguments hold for the finite-dimensional vector space of
Example C.2 since x, y =
n
i=1 x i y i , and the vector addition and multiplication
by scalars is performed component-by-component. As mentioned previously, the
square of the length of x is
n
i=1 x 2
i so that x, x = 0 if and only if x = 0.
Example C.6 The values of the elements f and g of the vector space of functions
in Example C.3 at arbitrary t can be viewed as “components” of these infinitedimensional vectors. The extension of Eq. C.6 to the continuous case yields the inner
product
f, g =
τ
0
f (t) g(t) dt.
(C.8)
That this definition satisfies the properties of the inner product follows from the
operations defined by Eq. C.5. Note also that the square of the “length” of f is
f, f =
τ
0 f (t) 2 dt so that it is zero if only if f (t) = 0, t ∈ [0, τ ].
Definition C.4 A set of vectors x r , r = 1, . . . , p, of R n is linearly independent if
p
r=1
α r x r = 0 implies α r = 0, r = 1, . . . , p,
(C.9)