Lecture #08

Scientific Computing Using Python - PHYS:4905

Lecture #08 - Prof. Kaaret

Vectors, linear operators, and vector spaces

These notes borrow from Linear Algebra by Cherney, Denton, Thomas, and Waldron.

What is a vector?

A physicist typically thinks of a vector as a way to specify the position of an object in a 3 or 2 dimensional space. There are also other vectors that are common in physics. For example, velocity can also be a vector, as can acceleration. What is common about these quantities?

One property common to all vectors is that they can be added and the result is another vector.

Another common property is that a vector can be multiplied by a number, that we'll call a scalar, and that the result is another vector. We'll call this scalar multiplication.

The concept of vector can be generalized. For example, instead of using a basis consisting of unit vectors $\{ \hat{i}, \hat{j}, \hat{k} \}$ , let's instead considering a basis consisting of the functions $\{ 1, x, x^2 \}$ defined over the interval $0 \le x \le 1$ . Instead of specifying a position in space, our vector $v = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ will now specify a function $a + bx + cx^2$ defined over that same interval.
Does this obey the two crucial properties of vectors defined above? Let's check. First, let's add two vectors,

$v + v' = \begin{pmatrix} a \\ b \\ c \end{pmatrix} + \begin{pmatrix} a' \\ b' \\ c' \end{pmatrix} = \begin{pmatrix} a+a' \\ b+b' \\ c+c' \end{pmatrix} = (a + bx + cx^2) + (a' + b'x + c'x^2) = (a + a') + (b+b')x + (c+c')x^2$
We can add the two functions specified by the two vectors and the resultant is another function of the same form. This means that we can add the two vectors and the resultant is another vector defined in the same basis. We can also multiply a vector by a scalar and the resultant is another vector. Therefore, the family of functions basis using the basis $\{ 1, x, x^2 \}$ works just like vectors representing position in a 3D space. Linear algebra is the study of vector spaces and how linear operators act on those spaces. By generalizing the concept of vector spaces, linear algebra becomes a very powerful mathematical tool.

Vector Spaces

We will be considered vectors with an arbitrarily large number of components: n-vectors.

We write them as

a = \begin{pmatrix} a^1 \\ a^2 \\ ⋮ \\ a^n \\ \end{pmatrix} \;\;\;\;\; or \;\;\;\;\; b = \begin{pmatrix} b^1 \\ b^2 \\ ⋮ \\ b^n \\ \end{pmatrix}

where the individual components are real numbers,

a^n ∊ ℝ

, where we use

ℝ

to denote the set of real numbers. Note that

a^2

means the second element of the vector a and not a squared. Also, note that in standard mathematical notion, one starts indexing from 1 rather than from zero as in Python.

Vectors exist in a 'vector space'. The vector space for an n-vector is the set of all possible n-vectors that particular n, which we can write as

{ℝ}^{n}

Euclidean Vector Spaces

We will initially be working in vector spaces with Euclidean geometry. This is the geometry that you are familiar with from 2 and 3 dimensional vectors in physics. Addition of vectors and multiplication of a scalar times a vector work as you expect

a + b = \begin{pmatrix} a^1 +b^1 \\ a^2 + b^2 \\ ⋮ \\ a^n + b^n \\ \end{pmatrix} \;\;\;\;\; and \;\;\;\;\; λa = \begin{pmatrix} λa^1 \\ λa^2 \\ ⋮ \\ λa^n \\ \end{pmatrix}

We define the dot product of two vectors as

a•b = a^1 b^1 + a^2 b^2 + ⋯ + a^n b^n

We define the Euclidean length of an n-vector as

\begin{Vmatrix} a \end{Vmatrix} = \sqrt{ (a^1)^2 + (a^2)^2 + ⋯ + (a^n)^2} = \sqrt{a•a}

and we define the angle between two vectors

θ

\begin{Vmatrix} a \end{Vmatrix} \begin{Vmatrix} b \end{Vmatrix} \cos(θ) = a•b

The dot product is

- commutative (or symmetric)

a•b = b•a

- distributive

a•(b+c) = a•b + a•c

- linear in both vectors or bilinear

a•(λb+λc) = λa•b + λa•c

- and positive definite

a•a ⩾ 0

This isn't the only possible way to define the dot product. If you've done any special relativity, you might have been introduced to a dot product that is not positive definite. In the Lorentzian inner product, the product of the terms in the time dimension comes into the sum with a minus sign. This is because space in relativity is not Euclidean.

Hyperplanes

In n-dimensional Euclidean geometry, we have a vector space

{ℝ}^{n}

that is full of points. We can use n-vectors to label particular points P. There is a special point, the origin, that we label with the 0 vector, which has all of its elements equal to zero. The zero vector is the only vector with zero length and no direction.

We can describe a line in

{ℝ}^{n}

in terms of two vectors, a and b, as the set of points

\{ a + λb | λ ∊ ℝ \}

We can describe a plane in

{ℝ}^{n}

in terms of three vectors, a, b, and c, as the set of points

\{ a + λb + μc | λ, μ ∊ ℝ \}

We can keep going and describe a hyperplane with k vectors a₁ ... a_k where

k ⩽ n

as the set of points

\left\{P + \underoverset{i=1}{k}{∑} λ_i a_i \, | \, λ_i ∊ ℝ \right\}

where we have replaced the vector pointing to a position with the point P at that position.

General Vector Spaces

The vector spaces

{ℝ}^{n}

are very nice vector spaces, but they are not the only possibilities. We could, for instance, consider the space of functions of one real variable. One such function is y = x, another is y = 3x², another is y = sin(2x). Each point in this space represents a function. We need an infinite number of numbers to specify every possible function, so the space is

{ℝ}^{ℝ}

. Note that the common operations that we use on vectors still work.

For example, we can add two functions f and g,

f(x) + g(x) = (f+g)(x)

Addition in this vector space means starting at one vector, adding another vector, and ending up at a final vector. It works just like vector additional in a Euclidean space, but the points in the space represent different functions.

This space also has a zero, defined as f(x) = 0. If we add the zero function to another function, we get back our original function. This is exactly the same as adding the zero vector.

Our more fancy vector spaces still need to follow a bunch of rules.

Closure under addition - adding two vectors in the space gives another vector in the space.
Addition is commutative - the order of addition doesn't matter.
Addition is associative - when adding multiple vectors, the order doesn't matter.
Zero - there is a zero such that adding it to any vector in the space leaves the vector unchanged.
Additive inverse - for any vector in the space, there is another vector such that the sum of the two vectors is zero.
Closure under scalar multiplication - multiplying any vector in the space by a scalar produces another vector in the space.
Multiplication is distributive over scalars and vectors.
Scalar multiplication is associative.
Unity - multiplying a vector times unity gives back the same vector.

We showed above that the space of functions of one real variable is a vector space. How about the space of differentiable functions of one real variable?

Is the sum of two differentiable functions another differentiable function?

Is the product of a scaler times a differentiable functions another differentiable function?

What is the zero of this vector space?

Fields (of Corn)

We have defined our vector spaces over the real numbers. In this context, the real numbers would be called the field or the base field or the baseball field, (well, maybe not that last one.) We could instead use a different field. In quantum mechanics, we use vector spaces to define the possible states of a physical system. For example, we might have an electron that can have its spin up, represented by

\begin{pmatrix} 1 \\ 0 \end{pmatrix}

, and its spin down, represented by

\begin{pmatrix} 0 \\ 1 \end{pmatrix}

. In a classical description of the electron, one would have some probability (a positive definite real number) that the electron is in the spin up state and a probability that the electron is in the spin down state. It would be nice if the two probabilities add to one, so that the electron is in one state or the other.

A very interesting aspect of quantum mechanics is that probabilities alone aren't good enough. We need to have a probability amplitude of the electron being in the spin up state and another probability amplitude that it is in the spin down state. One finds the probability on the electron being in a state by taking the modulus squared of the corresponding amplitude (and, again, it is nice if the probabilities add to one). However, complex amplitudes allow description of phenomena like interference.

Assignment

We're back to programming for HW #8.