Month: July 2023

Change of basis of a linear map

1. Coordinate maps

consider a random vector a. If the values of every entry are explicitly given, one could naturally identify it as a vector under the orthonormal basis. However, it could also been understood as the coordinate vector, which represents a vector under another basis by a coordinate map:

2. Change of basis of a linear map

Now we consider a linear map (represented by a matrix) A. It could be interpreted  as:

What if we want to change the basis of the objective vector space of f ? The following relation is obvious, if we want to find a matrix A’ representing the map after change of basis:

Where P is defined as a matrix that perform this basis change.

[Literature: Andre Lukas, Lecture note on Vectors and Matrices, University of Oxford]

3. The invariant map

Suppose we have a map that is invariant under any basis change, that is:

In other words, we would like to find an operator that commutes with any other operator on the same vector space V. Suppose now we have a vector x in V. We would like to find a non-trivial linear functional on x, so that we can define a linear map:

This is possible, for a functional in a n dimensional space can be express in to an (1 x n) matrix, so the left hand side can be expressed as (nx1)(1xn)(nx1) corresponding to v, f, and x respectively, and the former 2 combined and form a matrix. ( My reference mentioned ” Axiom of choice” with respect to finding non-trivial functional, and yet I have not understood it perfectly)

Then, the commutation relation implies:

Note that f(Tx), according to our pervious argument, should exist and independent on v. Thus, what we are doing consequently is that we have constructed linear maps P according to our need (that is, the vector v). In other words, we can assign every vector v in V a matrix P and they have to satisfy (6).

Then this inplies:

i.e. T is the scalar multiple of  the identity.

[Literature: Robert Isreal, https://math.stackexchange.com/questions/27808/a-linear-operator-commuting-with-all-such-operators-is-a-scalar-multiple-of-the]

 

Linear Functionals

1.Vectors

It is important to formalised the definition of a vector and a vector space. Here I shall pay no more attention on the this issue. But they are important as they characterised the condition that, for example, the scalar field accompanying with V, must satisfy.

1.Functionals

One should be very familiar with the definition of “linear” and “linear maps”(And from the definition we see linear maps can from a linear space: space of linear functions, as well). Very interestingly, we can show the following two important relations: for f: V→W:(See reference)

and that if ker= {0}, f: V→V is isomorphism.

We can get the following observation: consider a linear map with dim(imf) = 1 (Which, since vector spaces are defined base on a scalar field, this 1 dimensional space is the field itself. We name this special type of map linear functionals. Since we know that linear functions themselves form a vector space, we call it dual space V *(n,K) , this apply to linear functionals as well. Form (1) we know that one particular functional actually only act on one specific dimension, we conclude that there are only dim(V) types of linearly independent functionals. i.e. dim(V)=dim(V*). It follows that for basis {e} in V and basis {g} in V*, we naturally pair them together:

So that V and V* are putted in an equal status (symmetry).

Now, suppose we have f: V→W and g: W→F where g is a functional, it is easy to find the a functional in V*, namely, g( f ).  Then, we could also define another space of functionals h: W*→V* as h: g→f = h(g). 

We can also define a map i: V→V* and we can show the condition for it to be isomorphism. Since dimV = dim V* if we also know Ker(i) = 0 this should do it. Remember that for defining a dual space, it is necessary to recall that it represents a space of linear maps.  For any non-zero v, if w = f(v) is non zero (Note that the image of zero is always zero) the corresponding dual vector is (by our definition) non zero. (In other word, this map is not degenerate) However, we shall note that the proportionality between e and image of e by g is not restricted. We could try to more specify it:

Consider the effect on w, we also use it to define the inner product <,>: V x V→F:

[Reference: Geometry, Topology and Physics. M Nakahara(2003)]

Russell’s Paradox

1. The earliest from: Burali-Forti’s paradox

Though this form is only a subset of a general Russel’s paradox (from self reference), I have collected it for it involves an interesting concept in sets: The Ordinals.

We define a set to be an ordinal if it is hereditarily well-founded and hereditarily transitive. Hereditariness is defined as the specific characteristic of the set is inherited by every element of the set, and transitive is defined as the element of the element of a set is still the element of this set.

The direct observation of an ordinal O is that any element, sub element…etc. are all an element of O. Further more, any smallest indivisible element inside O is an element of O. Importantly, we immediately recognise that the set of all ordinals is, itself an ordinal (Let’s call it U ).

This property of ordinals make it possible to give well-defined orders in them. for example, consider an order type that rank each indivisible element, combined with the order type that rank the cardinality of sets (in terms of indivisible elements). In this way, U should be in a highest rank, higher than any element in U, includes itself.

2. Naive set theory and Russell’s paradox

The way that Russell spotted the paradox was through the unrestrictive comprehension axiom. This says that two formula f and g are identical iff f(x)= g(x) for any x. In this way we can construct a set {x: f(x)} to collect all x satisfying formula f(x).

Russell took f(x) as x is not an element of x. Then R= {x: f(x)} seems to be both an element and not an element of this set. This leads to a contradiction

3. Solutions to Russell’s paradox

Let us rewrite what Naive set theory gives us in a tidier mathematical form:

Russell’s response is call typed theory. This came from vicious circle principle stating that ” no proposition function can be defined in prior to specifying the function’s scope of application. Thus (1) should be modified:

Here S should be restrict x s.t. εf is not impredicative. R is defined in such a way that εf is impredictive so according to (2)  no such a set A exist.

Typed method, effectively, simple denies the set described by “all possible…., universal…” to be a set. However, there are other responses to the paradox. For instance, the axiom schema of Separation states that:

This is in similar manner to (3) but it does not directly state what set A should be. But now take f to be the one in Russell’s paradox gives the result of A does not exist.

Other arguments wanted to do it in different manner. Intuitionism and Para-consistency seems to be admit the existence of true contradiction and they altered some basic sentential logic. What are their relation to dialectics? I will come to that later.

[Reference: Stanford Encyclopedia of Philosophy: Russell’s Paradox]

 

Basic Thermodynamics

At the very, very beginning, let us see what thermodynamics up to: This is a science investigate the 1. To which forms energy goes to, and 2. how to make energy useful. Let us start from the first issue. It is a common-sense that energy consists of mainly to forms: (ability to do) Work and (ability to release) Heat. The former one corresponds to a directed, ordered moving, in contrast the latter shows a chaotic movements (in statistical physical sense). 

Thus we have the first law. It said that the change in total energy should be decomposed to heat and work. 

Since industrial revolution, the requirement of human on work rapidly increases, even (say) exceed the need on heat. This relates to the second objective of  thermodynamics: How to make energy useful. More specifically, we wanted to discover the convection laws better work and heat. 

To achieve that, we first need to identify what energies are useful. One may dream of convert thermal energy to work with infinitesimal cost, or in other word, convert an object’s thermal energy completely to work. However, this is not that useful  (and, in fact, impossible) : one need to “recycle” the row material , that is why we are interest in thermodynamical “cycles”. We wish for each cycle, the working substance can come back to its original “state” (For a state, we tried to find all what we called states variables that thermodynamically we cannot tell the difference between two substances with identical values of state variables). Our task is thus, identify the condition which a completely restored to the original state, while making useful effect, is possible. 

It is a common-sense that under some conditions our processes are dissipative, i.e. energy flow out regardless either direction we undergo in the process. Dissipative work includes friction, and dissipative heat includes heat lost. For instance, say we have a random process that involve (mechanical)work done. Let us further assume that the process involve at least one (either pressure of the object or the pressure of the surrounding) well-defined pressure. Whatever the process is, if originally (without friction) we expect the process can go back to the original state (both system and surrounding)  following the same path (We call this reversible), this time with friction we have the pressure deviated higher from the path in the forward process and deviated lower from the path  in the reverse process. This means it is no more reversible. Similarly, processes involving heat dissipation are again, irreversible. (We can show it quite generally)

There are other processes without dissipation that are irreversible. But as we have found out, reversible processes guaranteed non-dissipative nature, it means that we should be interested in these processes. Thanks to the study in heat engine by Carnot, we have a well-establish example of reversible processes: Carnot cycles. From the definition of Carnot cycle and another important statement: The second law of thermodynamics, one can derive the complete picture of basic thermodynamics, as we will discuss later.

One easiest approach to the second law is using microscopic point of view (Although is had not been understood by the physicists in the 19th century yet): Through a diffusion process, heat can indeed only spontaneously flow from hotter to colder objects, not reverse. This is Clausius’s statement of the second law. One can show that it is equivalent to the Kelvin-Plank’s statement. Using Clausius’s statement, one concludes that a reversible heat cycle operating between two reservoirs (apart from the trivial one: The one goes back to the starting point with identical path) should have the same efficiency as the Carnot’s cycle, which is naturally a maximum efficiency one can achieve as a heat engine. Reversible heat cycles as we defined, are thus indeed the most particular type of cycles (and in fact are fictitious) and are indeed the type we are seeking. 

Either using the efficiency relation (require the process is quasi static so that we can draw a path) or using the Kelvin’s statement, we can derive Clausius’s theorem for any closed cycle. On the other hand, it is easy to observe that any two points on the P-T-V diagram can be connected by reversible processes (If a states equation exist). Thus we can use Clausius Theorem to induce that for any two points on P-T-V diagram there exist a fixed quantity, if we set the quantity at one point is zero, then this quantity is well-defined in the entire space and independent on the path — from now on we call it entropy. 

Since entropy and  internal energy are both state variables, from the first law we conclude that for irreversible process, heat is always less than the corresponding reversible heat and thus work, must be greater than the corresponding reversible work.