16 – …But not as we know it

< ρℝεν | ℂσητεητs | ℕεχτ >

Warped space[see Acknowledgements for Image Credit]

“Beyond the edge of the world there’s a space where emptiness and substance neatly overlap, where past and future form a continuous, endless loop. And, hovering about, there are signs no one has ever read, chords no one has ever heard.”

– Haruki Murakami

 
In the previous Chapter we explored some properties of Cartesian vectors and determined that the structure which they formed was something called a Vector Space. This was rather a loose and informal way of proceeding (though hopefully none the less illuminating). Here my aim is to be more rigorous and indeed more general. Vector Spaces are not just collections of arrows, but broader Mathematical entities. While they may share some properties, many Vector Spaces are very different to our traditional concept of what constitutes a vector. To put it another way, many vectors are not a line with direction and magnitude.

Before moving on to offer a formal definition of a Vector Space, there is a loose end to deal with. In the previous Chapter I have rather ambiguously referred to scalars as numbers. Well of course they are numbers, but of what kind? The answer is that, in order for the overall structure to be a Vector Space, the scalars have to be part of another Mathematical construct known as a Field [1]. Fields are an important concept in Mathematics in their own right and the next box explains what a Field is. If you are happy with the formulation “Field = some sort of numbers” as I have used previously, then feel free to skip this box.

 
Pastures New

John Constable - West End fields [see Acknowledgements for Image Credit]

So I wasn’t entirely out of order when I referred to scalars like λ and μ being numbers. If I had said either Rational Numbers (ℚ) or Real Numbers (ℝ) then I would have been closer to the truth, while also noting that if I had instead said Natural Numbers (ℕ) or Integers (ℤ), I would have been misleading. What is important about the scalars chosen as part of a Vector Space is that they support addition and multiplication, but also division. E.g. if the set of scalars is called F, then if a, b ∈ F we also have a/b ∈ F (assuming b is not the identity under addition, aka zero in most circumstances). Here we can see why ℚ and ℝ will work but neither ℕ nor ℤ will fit the bill.

A Mathematical structure that behaves the same way as ℚ or ℝ with respect to addition, multiplication and division is called a Field. There are some quite recherché examples, but if you think of ℚ or ℝ as the prototype, then you won’t go far wrong. The set of formal properties that a Field exhibits is actually not that different to a Vector Space in some respects. For a Field F, they are as follows:

  1. Closure

    For all a, b ∈ F:

    a + b ∈ F and

    ab ∈ F
     

  2. Commutativity

    For all a, b ∈ F:

    a + b = b + a and

    ab = ba
     

  3. Identities

    There exist 0 and 1 ∈ F, such that for all a ∈ F:

    a + 0 = a and

    1a = a
     

  4. Inverses

    For all a ∈ F, there exists -a ∈ F, such that:

    a + -a = 0.

    For all a ∈ F where a ≠ 0, there exists a-1 ∈ F, such that:

    aa-1 = 1
     

  5. Associativity

    For all a, b, c ∈ F:

    a + (b + c) = (a + b) + c and

    a(bc) = (ab)c
     

  6. Distributivity

    For all a, b, c ∈ F:

    a(b + c) = ab + ac
     

From the above we can see that a Field F is effectively two for the price of one with respect to Groups, (F,+) is a Group and so is (F\0,×) [2]. What is more both Groups are Abelian.
 

 
 
The Nature of Space (not Time)

Hubble image of a star field [see Acknowledgements for Image Credit]

So the reader either now knows what a Field is, or is willing to accept my earlier definition of “some numbers”. Either way, we are now equipped to offer a formal definition of a Vector Space as follows:

A Vector Space, consists of two structures, a set of vectors, V, and a set of scalars, F, where F is a Field. On this are defined two binary operations. Vector addition (V × V ↦ V) takes any two members of V, say v and u and combines then to create a further element of V, v + u. Scalar multiplication (F × V ↦ V) takes any element of V, say v, and any element of F, say f, and combines them to create another element of V, fv.

The above definition implies already that the combination of V and F is closed (because v + u ∈ V and fv ∈ V). In addition the Vector Space must possess the following properties:

  1. Additive Identity

    There exists a vector 0 ∈ V, such that for all v ∈ V:

    0 + v = v
     

  2. Additive Inverses

    For all v ∈ V, there exists –v ∈ V such that:

    v + –v = 0
     

  3. Additive Associativity

    For all u, v, w ∈ V:

    u + (v + w) = (u + v) + w
     

  4. Additive Commutativity

    For all u, v ∈ V:

    u + v = v + u
     

  5. Compatibility of Addition and Scalar Multiplication

    For all f, g ∈ F and all v ∈ V:

    g(fv) = (gf)v [3]
     

  6. Scalar Identity

    There exists a number 1 ∈ F such that, for all v ∈ V:

    1v = v
     

  7. Distribution of Scalars over Vector Addition

    For all f ∈ F and u, v ∈ V:

    f(u + v) = fu + fv
     

  8. Distribution of Scalar Addition over Scalar Multiplication

    For all f, g ∈ F and v ∈ V:

    (f + g)v = fv + gv

That’s a lot of properties I agree, ten if you include the two implied Closure properties. Once more Vector Spaces are an important concept in and of themselves, but to help us continue our journey towards SU(3), we want to consider a special type of Vector Space. This will be covered in the next Chapter. I will close this Chapter by introducing another facet of Vector Spaces, one which we will employ again soon, a basis of a Vector Space.
 
 
Back to Bases

I was 21 years when I found this set, I’m 22 now, and I'm glad that we met [see Acknowledgements for Image Credit]

To kick off this section, let’s again consider the diagram I used at the beginning of the Chapter, the one with the red arrow on the graph paper. In this the vector, v is decomposed into x and y component vectors, vx and vy respectively. If we now consider the situation where v is bound to the origin, (0, 0), we can see that both vx and vy would also be bound to the origin; this is just moving the rectangle that vx and vy form two sides of (with v as its diagonal) until the bottom left-hand corner coincides with (0, 0).

From other Chapters the reader may recall that notation that |x| defines “size of x”. This concept also applies to at least this type of vector, |v| is just the length of the arrow, a non-directional, or scalar quantity. Using this notation we can see that:

|vx| = x-coordinate of v and

|vy| = y-coordinate of v

So we can also see that:

v = (|vx|, |vy|)

Which is of course another way of stating that:

v = vx + vy

Because:

vx = (|vx|,0)

and

vy = (0, |vy|)

In the above, I have totally ignored the fact that v = (18, 13) and in doing so have shown that any vector in this particular Vector Space can be defined by adding together its x- and y-coordinates. We could go further of course.

Let us consider the two vectors:

1x = (1, 0) and

1y = (0, 1)

We can visualise these as follows:

2D basis

Although I never made this explicit before, if we assume that our “graph paper” is gradated using the Real Numbers, ℝ (i.e. ℝ forms the Field of the Vector Space V2D), then if we pick any two numbers a and b ∈ ℝ, we can construct a vector v(a,b) as follows:

v(a,b) = a1x + b1y

= a(1, 0) + b(0, 1) = (a, 0) + (0, b) = (a, b)

We can see that by choosing a and b appropriately, we can construct any vector in V2D. Again if we look at this visually we get:

2D basis example

Turning this around, it is a long-winded way of stating what mighty now appear to be blindingly obvious. Any vector in V2D can be constructed by picking appropriate scalars a and b, multiplying our two vectors 1x and 1y by these in turn and adding the result.

This is the essence of a basis [4] of the Vector Space V2D, a set of two vectors, {1x, 1y}, which when combined and multiplied by appropriate scalars can generate any member of the Vector Space. We say that the basis spans the Vector Space.

I have picked unit vectors (ones whose length is one) for my basis, but it should be noted that bases are not unique. Instead of {(1, 0), (0, 1)}, I could have picked {(22, 0), (0, 13)} instead. If with the original choice of basis I had to use a and b to create a given vector, instead I now use a/22 and b/13. In practice unit vectors tend to be more convenient for obvious reasons.

However, it is not only the size of our two basis vectors that can be different. We don’t need the angle between them to be 90° either. If I instead picked a set with members {(1/√2, 1/√2), (0, 1)}, both of which are still unit vectors, then this would look like:

2D basis 45 degrees

Although it would be more fiddly (and I don’t propose to demonstrate this here), we could still use this to create any vector in V2D by choosing appropriate scalars. What is important is that you can use neither of these two new vectors to create the other one, that is:

For all a ∈ ℝ

a(1/√2, 1/√2) ≠ (0, 1)

This inability to create one vector from another is called linear independence.

Here we have been considering a 2D Vector Space, V2D. If we instead consider a 3D Vector Space, such as V3D, which we defined above, then unsurprisingly the set {(0, 0, 1), (0, 1, 0), (1, 0, 0)}, forms the most natural basis. If we use the same notation as before for the first two vectors and 1z for third one, then here the requirement for the basis elements to be linearly independent becomes:

For all a, b ∈ ℝ

a1z + b1y1z

Which we can clearly see is true for these three vectors.

Here again each of 1x, 1y and 1z are both unit vectors and mutually at right angles to each other (orthogonal in the parlance). Once again vectors in a basis for V3D don’t need to be either of length one or orthogonal. The only requirement is that they be linearly independent of each other. Indeed it may be shown that:

For a Vector Space [5], Vn, of dimension n, over a field F, any set of n linearly independent vectors, {v1, v2, … , vn}, form a basis for Vn. That is, for any v ∈ Vn, we can find n elements of F, {f1, f2, …, fn}, such that:

v = f1v1 + f2v2 + … + fnvn [6]

Because I have been using the geometric example of Cartesian Vector Spaces, the idea of dimension falls out of this naturally; it is the number of axes, or equivalently the number of coordinates of each vector. In the case of a more generalised Vector Space, its dimension is the size of any of any set forming a basis for it [7].
 
 
Exempli Locis

Exempli Locis [see Acknowledgements for Image Credit]

Nothing clarifies a rather recherché definition like a few bracing examples. Having spent a lot of time talking about arrows in the last Chapter, here are a few Vector Spaces which are less arrow-like.
 
 
1. Complex Numbers

Back in Chapter 7 we introduced the set of Complex Numbers denoted by ℂ. Member sof ℂ are of the form a + ib, where a, b ∈ ℝ. We can think of Complex Numbers as being our vectors and, given our definition of ℂ, our obvious candidate for a Field is ℝ.

Back in Chapter 7 we mentioned (without proving) that the Complex Numbers are an Abelian Group under addition, something that is fairly obvious to see from the rule that:

(a + ib) + (c + id) = (a + c) + i(b + d)

If we define scalar multiplication in the obvious manner:

α(a + ib) = αa + iαb

it can be seen that the other requirements for ℂ to be a Vector Space over ℝ are fulfilled.

The obvious choice of basis is the set {1 + i0, 0 + i0} or just {1, i}. We can see that these are linerarly independent as there is no a ∈ ℝ such that ai = 1. So the dimension of ℂ as a Vector Space is 2.

In Chapter 7 we used the concept of the Complex Plane to visualise Complex Numbers. If we think about this further it is probably relatively easy to see that the two Vector Spaces ℂ and V2D are isomorphic [8] with the mapping, φ, between them being:

φ: ℂ ↦ V2D, a + ib ↦ (a, b)

So in a sense ℂ and V2D are the same Mathematical structure.
 
 
2. Polynomials

We met polynomials back earlier in the book and defined them in Chapter 11. They are expressions of the form:

anxn + an-1xn-1 + … + a2x2 + a1x + a0

Where the ai, with i ranging from 0 to n, are called the coefficients and n is known as the degree of the polynomial.

Let us consider the set of all polynomials of degree less than or equal to n [9]. We will consider each polynomial as being a vector with the coefficients as being members of the Real Numbers, ℝ. Perhaps unsurprisingly we will also use ℝ as our Field.

It is first of all worth noting that two polynomials are equal only if their coefficients are equal and vice versa, i.e.:

anxn + an-1xn-1 + … + a2x2 + a1x + a0 =

bnxn + bn-1xn-1 + … + b2x2 + b1x + b0

Only if ai = bi for 0 ≤ i ≤ n and vice versa.

Vector addition is then defined as:

anxn + an-1xn-1 + … + a2x2 + a1x + a0 +

bnxn + bn-1xn-1 + … + b2x2 + b1x + b0 =

(an + bn)xn + (an-1 + bn-1)xn-1 + … + (a2 + b2)x2 + (a1 + b1)x + (a0 + b0)

And scalar multiplication, is defined as:

α(anxn + an-1xn-1 + … + a2x2 + a1x + a0) =

αanxn + αan-1xn-1 + … + αa2x2 + αa1x + αa0

It can be seen from the above that both vector addition and scalar multiplication are localised to adding individual coefficients or multiplying individual coefficients by scalars. Without working through all of the properties that need to be fulfilled for these polynomials to be a Vector Space, we can see that they will follow how the coefficients are combined. As the Field is ℝ and the coefficients are also in ℝ, it is not a stretch to see that this set is also a Vector Space.

An obvious basis also suggests itself, the set of polynomials:

{1, x, x2, … ,xn-1, xn}

or written longhand:

{0xn + 0xn-1 + … + 02x2 + 01x + 1,

0xn + 0xn-1 + … + 02x2 + 11x + 0,

0xn + 1xn-1 + … + 02x2 + 01x + 0,

1xn + 0xn-1 + … + 02x2 + 01x + 0}

This means that the dimension of Pn as a Vector Space is actually n +1 (count the elements of the basis). We will come back to this pattern of 1s and 0s later.

Once more we can also envisage an isomorphism from the set of polynomials of degree less than or equal to n (let’s call this Pn) to V(n+1)D, ψ as follows:

ψ: Pn ↦ V(n+1)D, anxn + an-1xn-1 + … + a2x2 + a1x + a0 ↦ (an, an-1, … , a2, a1, a0)

Once more we see a connection between ostensibly discrete branches of Mathematics. And incidentally we have also shown that (Pn, +) is a Group, Groups can appear in the most surprising places.
 
 
3. 3 × 3 Matrices

Vector Spaces comprised of matrices are things we are going to see more of in the next Chapter, so it is probably worth taking some time to understand these in more detail. I’m generally not intrepid enough to deal with anything bigger than 2 × 2 matrices, but let’s make an exception here and consider the next step on the rung [9]. As we saw in Chapter 5, where the entries in the matrices are Real Numbers this set can be denoted by M3(ℝ) [11]. Let’s say our vectors are members of M3(ℝ) and make the obvious choice of our Field as the Real Numbers.

We then have vector addition being normal matrix addition and scalar multiplication being normal scalar matrix multiplication as follows:

3x3 vector addition and scalar multiplication

Where aij, bij, λ ∈ ℝ.

From Chapter 5 we already know that Mn(ℝ) forms a Group under matrix addition [12], which gets us part of the way to showing that it is a Vector Space as well. Again because the definitions result in vector addition and scalar multiplication consisting of performing operations to each cell of the matrices separately and the members of each cell are part of a Field, it follows that Mn(ℝ) is indeed a Vector Space.

Once more an obvious basis presents itself:

3x3 basis

From this we can see that the dimension of M3(ℝ) is 9.

At first it might seem that we would struggle to map our 3 × 3 matrices to a coordinate Vector Space as we did for both ℂ and Pn, however consider a function, χ, as follows:

3x3 isomorphism

If we consider each of the rows of our 3 × 3 matrix as being a vector, then we have formed one bigger vector by lying these side by side.
 
 

Note: At this point I’m going to drop my own notation of VnD(ℝ) for the coordinate space of degree n over the Real Numbers.

If instead of ℝ we consider some more generic Field, F, and use this to populate n-dimensional vectors (f1, f1, …, fn) where fi ∈ F, then the typical notation is Fn.

So above we established an isomorphism from M3(ℝ) in to ℝ9.

 
 
Having provided a more general definition of a Vector Space and briefly explored some perhaps less obvious examples, in the Chapter after next we will spend a lot of time looking at a specific type of Vector Space. These Vector Spaces will also possess some additional attributes which are very pertinent to the subject of this book. These are called Lie Algebras.

However, before we do this, there are some other properties of matrices that we need to explore, ones with both Mathematical and Physical resonances. The properties are Eigenvectors and Eigenvalues. These will be the subject of the next Chapter.
 
 

Concepts Introduced in this Chapter
Field A Mathematical structure which supports two binary operators akin to addition and multiplication and which behaves like the Rational Numbers or Real Numbers; i.e. it also supports division and addition and multiplication can be combined in a sensible manner. If our Field is called F and our binary operators are denoted by the customary symbols, then F’s properties may be summarised by saying that (F,+) is an Abelian Group and so is (F\0,×) (where 0 is the identity element of the additive Group).
Vector Space A generalisation of the concept of a set of arrow-like vectors. A Vector Space is defined with respect to a Field F and supports addition of vectors and multiplication of vectors by scalars drawn from F. Again if our Vector Space is denoted by V, the (V,+) is an Abelian Group and vector addition and scalar multiplication may be combined in sensible ways.
Basis of a Vector Space In V is a Vector Space over a field F, a basis for V is a set of elements {v1, v2, … , vn} ∈ V, such that for any other v ∈ V, there exist {f1, f2, … , fn} ∈ F, such that:

v = f1v1 + f2v2 + … + fnvn

The basis is said to span V. The dimension of a Vector Space is defined as the number of elements in a basis for it (all bases of a Vector Space have the same size)

Linear Independence A set of vectors {v1, v2, … , vn} in a Vector Space V over a Field F are said to be linearly independent if, for all f1, f2, … , fn ∈ F:

f1v1 ≠ f2v2 + … + fnvn

That is no vector in the subset can be expressed in terms of the other vectors. For a Vector Space V of dimension n, any set of n linearly independent vectors forms a basis.

Groups Discovered in this Chapter
Pn The set of all polynomials of degree less than or equal to n under polynomial addition, i.e.

{anxn + an-1xn-1 + … + a2x2 + a1x + a0, +}

In our example we had ai ∈ ℝ but the same result would be achieved with the coefficients belonging to any set of numbers which are themselves a Group with respect to addition, so each of ℕ, ℚ and ℂ would work, along with some more esoteric constructions.

< ρℝεν | ℂσητεητs | ℕεχτ >

Chapter 16 – Notes

 
[1]
 
Mathematics – or at least Pure Mathematics – is sometimes described as the discipline that uses simple words for complicated concepts. We have already met a few of these along the way including: Group, Simple, Normal, Function, Prime, Rational, Real, Imaginary and now Field. Other examples of everyday words that mean something totally different in Mathematics would include Ring, Graph (not a line drawn of the criss-cross lines I use at the beginning of this Chapter) and Ideal; there are many more.
 
[2]
 
We need to remove the additive identity from F in order for it to be a Group under multiplication, the same as we did for both ℚ and ℝ back in Chapter 4.
 
[3]
 
g(fv) implies scalar multiplication of v by f and then scalar multiplication of the resulting vector by g.

(gf)v implies multiplication in the Field F of g and f with the result used as a scalar multiple of v.

 
[4]
 
Another of our complicated ideas hiding behind a simple word of course.
 
[5]
 
An object adhering to the general definition I presented above.
 
[6]
 
Two points here. You can also have Vector Spaces of infinite dimension and so a basis for them will have infinite members (by definition). In this case the requirement is that any member of the Vector Space must be generated from a finite sum of basis elements, each multiplied by scalars.

Also the reader might question whether or not all Vector Spaces must have bases. Perhaps surprisingly, the existence (or otherwise) of bases for all Vector Spaces is wholly equivalent to The Axiom of Choice; something that is beyond this scope of this book to cover, but which is again another example of the deep connections between different areas of Mathematics.

 
[7]
 
All bases of a Vector Space will be of the same size, again I don’t propose to prove this here.
 
[8]
 
We introduced the concept of Isomorphism with respect to Groups in Chapter 13. It works pretty much the same way for Vector Spaces, which are in any case Groups from the point of view of the additive operator.
 
[9]
 
If n=4 for example, we can write a polynomial of degree 3 as:

0x4 + a3x3 + a2x2 + a1x + a0

So we can assume all polynomials of degree less than or equal to degree 4 can be written as polynomials of degree 4.

 
[10]
 
The comments of course apply to n × n matrices as well.
 
[11]
 
Mn(F) is also a Vector Space, where F is any Field.
 
[12]
 
It is worth recalling that under matrix multiplication it is not Mn(ℝ) that forms a Group but rather its subset GLn(ℝ) the n × n invertible matrices, more on this in later Chapters.

Text: © Peter James Thomas 2016-17.
Images: © Peter James Thomas 2016-17, unless stated otherwise.
Published under a Creative Commons Attribution 4.0 International License.