1Vector spaces
IB Methods
1 Vector spaces
When dealing with functions and differential equations, we will often think of
the space of functions as a vector space. In many cases, we will try to find a
“basis” for our space of functions, and expand our functions in terms of the basis.
Under different situations, we would want to use a different basis for our space.
For example, when dealing with periodic functions, we will want to pick basis
elements that are themselves periodic. In other situations, these basis elements
would not be that helpful.
A familiar example would be the Taylor series, where we try to approximate
a function f by
f(x) =
∞
X
n=0
f
(n)
(0)
n!
x
n
.
Here we are thinking of
{x
n
:
n ∈ N}
as the basis of our space, and trying to
approximate an arbitrary function as a sum of the basis elements. When writing
the function
f
as a sum like this, it is of course important to consider whether
the sum converges, and when it does, whether it actually converges back to f.
Another issue of concern is if we have a general set of basis functions
{y
n
}
,
how can we find the coefficients
c
n
such that
f
(
x
) =
P
c
n
y
n
(
x
)? This is the
bit where linear algebra comes in. Finding these coefficients is something we
understand well in linear algebra, and we will attempt to borrow the results and
apply them to our space of functions.
Another concept we would want to borrow is eigenvalues and eigenfunctions,
as well as self-adjoint (“Hermitian”) operators. As we go along the course, we
will see some close connections between functions and vector spaces, and we can
often get inspirations from linear algebra.
Of course, there is no guarantee that the results from linear algebra would
apply directly, since most of our linear algebra results was about finite basis and
finite linear sums. However, it is often a good starting point, and usually works
when dealing with sufficiently nice functions.
We start with some preliminary definitions, which should be familiar from
IA Vectors and Matrices and/or IB Linear Algebra.
Definition
(Vector space)
.
A vector space over
C
(or
R
) is a set
V
with an
operation + which obeys
(i) u + v = v + u (commutativity)
(ii) (u + v) + w = u + (v + w) (associativity)
(iii) There is some 0 ∈ V such that 0 + u = u for all u (identity)
We can also multiply vectors by a scalars λ ∈ C, which satisfies
(i) λ(µv) = (λµ)v (associativity)
(ii) λ(u + v) = λu + λv (distributivity in V )
(iii) (λ + µ)u = λu + λv (distributivity in C)
(iv) 1v = v (identity)
Often, we wouldn’t have just a vector space. We usually give them some
additional structure, such as an inner product.
Definition
(Inner product)
.
An inner product on
V
is a map (
·, ·
) :
V ×V → C
that satisfies
(i) (u, λv) = λ(u, v) (linearity in second argument)
(ii) (u, v + w) = (u, v) + (u, w) (additivity)
(iii) (u, v) = (v, u)
∗
(conjugate symmetry)
(iv) (u, u) ≥ 0, with equality iff u = 0 (positivity)
Note that the positivity condition makes sense since conjugate symmetry entails
that (u, u) ∈ R.
The inner product in turn defines a norm
kuk
=
p
(u, u)
that provides the
notion of length and distance.
It is important to note that we only have linearity in the second argument.
For the first argument, we have (λu, v) = (v, λu)
∗
= λ
∗
(v, u)
∗
= λ
∗
(u, v).
Definition
(Basis)
.
A set of vectors
{v
1
, v
2
, ··· , v
n
}
form a basis of
V
iff any
u ∈ V can be uniquely written as a linear combination
u =
n
X
i=1
λ
i
v
i
for some scalars
λ
i
. The dimension of a vector space is the number of basis
vectors in its basis.
A basis is orthogonal (with respect to the inner product) if (
v
i
, v
j
) = 0
whenever i 6= j.
A basis is orthonormal (with respect to the inner product) if it is orthogonal
and (v
i
, v
i
) = 1 for all i.
Orthonormal bases are the nice bases, and these are what we want to work
with.
Given an orthonormal basis, we can use the inner product to find the
expansion of any u ∈ V in terms of the basis, for if
u =
X
i
λ
i
v
i
,
taking the inner product with v
j
gives
(v
j
, u) =
v
j
,
X
i
λ
i
v
i
!
=
X
i
λ
i
(v
j
, v
i
) = λ
j
,
using additivity and linearity. Hence we get the general formula
λ
i
= (v
i
, u).
We have seen all these so far in IA Vectors and Matrices, where a vector is a list
of finitely many numbers. However, functions can also be thought of as elements
of an (infinite dimensional) vector space.
Suppose we have
f, g
: Ω
→ C
. Then we can define the sum
f
+
g
by
(f + g)(x) = f(x) + g(x). Given scalar λ, we can also define (λf )(x) = λf (x).
This also makes intuitive sense. We can simply view a functions as a list
of numbers, where we list out the values of
f
at each point. The list could be
infinite, but a list nonetheless.
Most of the time, we don’t want to look at the set of all functions. That
would be too huge and uninteresting. A natural class of functions to consider
would be the set of solutions to some particular differential solution. However,
this doesn’t always work. For this class to actually be a vector space, the sum
of two solutions and the scalar multiple of a solution must also be a solution.
This is exactly the requirement that the differential equation is linear. Hence,
the set of solutions to a linear differential equation would form a vector space.
Linearity pops up again.
Now what about the inner product? A natural definition is
(f, g) =
Z
Σ
f(x)
∗
g(x) dµ,
where
µ
is some measure. For example, we could integrate d
x
, or d
x
2
. This
measure specifies how much weighting we give to each point x.
Why does this definition make sense? Recall that the usual inner product
on finite-dimensional vector spaces is
P
v
∗
i
w
i
. Here we are just summing the
different components of
v
and
w
. We just said we can think of the function
f
as
a list of all its values, and this integral is just the sum of all components of
f
and g.
Example. Let Σ = [a, b]. Then we could take
(f, g) =
Z
b
a
f(x)
∗
g(x) dx.
Alternatively, let Σ = D
2
⊆ R
2
be the unit disk. Then we could have
(f, g) =
Z
1
0
Z
2π
0
f(r, θ)
∗
g(r, θ) dθ r dr
Note that we were careful and said that d
µ
is “some measure”. Here we are
integrating against d
θ r
d
r
. We will later see cases where this can be even more
complicated.
If Σ has a boundary, we will often want to restrict our functions to take
particular values on the boundary, known as boundary conditions. Often, we
want the boundary conditions to preserve linearity. We call these nice boundary
conditions homogeneous conditions.
Definition
(Homogeneous boundary conditions)
.
A boundary condition is
homogeneous if whenever
f
and
g
satisfy the boundary conditions, then so does
λf + µg for any λ, µ ∈ C (or R).
Example.
Let Σ = [
a, b
]. We could require that
f
(
a
) + 7
f
0
(
b
) = 0, or maybe
f
(
a
) + 3
f
00
(
a
) = 0. These are examples of homogeneous boundary conditions.
On the other hand, the requirement f (a) = 1 is not homogeneous.