note on notation and choice of indices

examples of Poisson-Bracket notation: 
  *  [(d)]
  *  [(f)]
  * note how version (f) opts for supercripted indices for one set of canonical
  coords. and subscripted for the other, while the (d) equivalent chooses 
  the simpler notation with all indices subscripted.
  * In [Siegel] [Srednicki] [Aitchison] and others there are discussions on the 
  importance of the choice of indexing scheme used in the QFT commutator formalism 
  - which is already so complicated (;;) .
  * Notation in another version (r) uses square brackets for the PB on the LHS,
  eg. [f,g] , thus making it indistinguishable from the notation used to designate
  the related quantum commutator, the Lie Bracket [a,b]. (Cf. introduction of the 
  commutator in [Shankar])
  * Indeed in __Fields__, Siegel devotes §§A,B in the Symmetry chapter to 
  "Coordinates" and "Indices" resp. [Siegel]

-- 24.xi.2008

linearity ≅ whole=∑parts?

In general, the main condition or step in proving a set to be a linear space is to show that for any two members f, g of the set under consideration, and for any two reals α,β

a. the linear combination αf + βg is in the set
b. this satisfies closure, from whence follow the remaining axioms 
  if applicable.

As an illusration, consider an operator qcq , L: D → ℝ^ℝ , where D is some set of functions , and ℝ^ℝ is the set of all real valued functions of a real variable.

L is linear, ie, is a linear space if , ∀y∈D, ∀z∈D, ∀α∈ℝ , ∀β∈ℝ

L(αy+βz)= αL(y)+βL(z)

This suggests some meaning. Viz., that operations on / properties of the whole are equal to sums of operations on / props of the parts.

This property appears to be akin to that of self-similarity, suggesting a rather profound meaning for the character of natural organization.

The ubiquity of linearity in the abstract algebra by which we represent natural systems^** (as noted earlier) has such consequences as the integrable character of physical law (discussed elsewhere).

^** if theory were complete then one could say up to an isomorphism, ie, the mathematical formalism is in a one-to-one, onto and domain-covering image of the fields of the physical system being described.

baby group - reading schiller

Part of the skill of the craftsmen behind familiar patterns such as this one snipped from Schiller's sprawling FREE book Motion Mountain,





is that the pattern of interwoven (wicker-like) white lines hides the much simpler structure of the underlying patches of color that have well defined shapes. But already by reducing the size of the figure, the color regions become clearer, and so does the simpler underlying design.


To facilitate identifying the color patches , we simplify the pattern by removing the white lines (roughly) and exposing the pattern that was hidden under their clutter.

The colored patches now become clear:





Now we can consider the transformations on this figure that Schiller talks about, and which were unclear before.

The following is largely a distillation of Schiller's discussion .

One of the symmetry transformations that leave the pattern unchanged is rotations in its plane about the origin.

The pattern is unchanged under rotations of pi/2.


It has only four positions in which it appears identical.

In general transformations on a square that leave it looking the same form a group, called dihedral.

Any transformation or bunch of transformations that leave an object appearing unchanged is called a symmetry.

Any collection of symmetries forms a group, a symmetry group.

For a square we can enumerate the possible such symmtery transformations:

rotations and reflections (flips).
We can rotate a square by &plusminus; 90 180 and 270 degrees , leaving it unchanged.
We reflect a square about either diagonal (flip it diagonally)
or reflect the sq. horizontally or vertically (flip it hor. or vertically) and it remains unchanged.

Along with counting the rotation by 0 degree, which is the equivalent of no rotation, which is called an Identity transformation (meaning no-op or do-nothing transformation) , we thus have 8 possible transformations.

To repeat, These form a group called dihedral, hence Schiller refers to it by its symbol as group D₄.

Though Schiller says there is only one transformation - rotation about pi - I don't see it . What i see is that
we have rotations about increments of pi/2 not pi. in other words , yes the figure looks the same when you rotate it 180 degrees,
but it will also look the same when you rotate it ninety degrees. So here either I misunderstand Schiller (more likely)
or something gives.

Apparently he's considering a reflection about pi. yet his group representations use cos n pi/2 and sin n p/2 .

also weirdly he cites a reflection matrix rather than a rotation matrix, unless on (p. 202) he uses a rotation matrix.


With every rotation transformation applied to the figure,
we can see that there are sets of shapes that get transformed into each other with every transformation.

Each set of these identical shapes that get transformed into one another with each transformation is called a multiplet.

To see this, we have the same figure with the multiplet sets enumerated. Again these are the sets of similarly shaped objects that transform into each other under all symmetry transformations (those that leave the properties of the object invariant - here it is the pattern, or the shape - ) such as rotation.

the numbered areas belong each to a multiplet indicated by the number.

Some of the multiplet sets are numbered, with the elements of each set given the number of that set.

Some multiplets are not numbered, but have arrows pointing at them.

Each multiplet as a whole (its entire set of elements) has the same symmetry as the overall figure.

For some of the colored shapes, the multiplet needs four objects to make up a whole multiplet (e.g.,
in the case of multiplet numbered 1 , we have identified its four elements).

Another multiplet, number 5, looks like it has 8 elements, but I am guessing since the symmetry degree of the multiplets is the same as that of the group.

Schiller points out another multiplet that has only one member, the central star. It is a one-element multiplet.

In any symmetry system such as this arabesque pattern, each part or component of the system is "classified" by the type of multiplet it belongs to.

Multiplets are also known as group representations.

More formally than the kindergarten chitchat above,

the representation of a group (aka symmetry group) is

an assignment of linear transformation matrix A(g) to each group element s.t.
A(g ∘ h) = A(g) ∘ A(h)

More formally still it is a homomorphism from the space of linear transformations onto the elements of the symmetry group.

Representations of unitary transformations are called unitary. Unitary transformations are matrices s.t. A^*=A^-1.

These have eigenvalues of norm (value?) 1 (perhaps leading to principle of gauge invariance? - afterall a gauge is a seminorm, and since norm is always one, gauge is invariant?) and mappings of unitary matrix are one-to-one.


If a matrix is not 121 it is singular, its determinant is zero, and it has no inverse transformation.

Almost all representations appearing in physics are unitary.

time evolution of physical systems is always described by the always one-to-one unitary mappings because these
map from time t-1 to t in a one-to-one fashion.

All unitary representations correspond to transformations that are one-to-one and invertible (both properties ,
equivalent to saying that determinant of the matrix is not zero and the matrix is thus non-singular.

A matrix whose det is zero is called singular as it has no inverse matrix. It cannot thus belong to a group.

Schiller then discusses reducibility of representations , identifying submultiplets , and giving rise to a classification of representations.

E.g., the pattern's symmetry group (called approximate) "has eight elements.

It has the general faithful unitary and irreducible group representations." and is an octet

Giving rise to the notation D₄(The so-called "approximate symmetry group") in the given figure, denoted D_4 , has eight elemnts.)

In any case it is from the symmetry that the deduction of the "list of multiplets or representations" that describe the symmetry's "building blocks" is possible.

Apparently other symmetry groups are given for other multipets? ie the representations are reducible to singlets doublets and quartets? This is what Schiller seems to say.

Schiller notes how unlike the transformations of the tiling pattern , which were discrete, the viewpoint transformations under which the world demeures unchanged are continuous and unbounded.

Siegel's Fields also states that continuous symmetry "is one of the most fundamental and important concepts of physics."

Continuity of the xforms means their representations are continually variable, without bounds and notably are magnitudes. In other words, scalars - as opposed to vectors. They can only be scalars.

Schiller notes ominously that in contrast vectors and tensors "only scalars may take discrete values", "may be discrete observables." (p. 204)

But weren't representations just made continually variable b/c nature's symmetry transformations are continuous? sounds a bit coucou .

To make things worse Schiller states that most representations also possess direction.
So they're not scalars anymore ?
Also it is states that symmetry under change of observ. pos inst or orientation => all observables are scalars, vectors, tensors or spinors (in asc. ord. of generality).









BTW rotations form (i.e., are) an Abelian group because g ∘ h = h ∘ g.

And generally it seems that a group has higher symmetry than its subgroups; iow, a group is a "larger symmetry group" than its subgroups. Makeosa della sensa.



* * *


The b&w lines thrown on top of the layer of multiplets (the color patches or geometrically shaped regions) (that get mapped to each other when undergoing transformations like rotation) are not mere spaghetti thrown on top of the rotation group representations.

That is to say, the layer of interlocking b&w lines is not trivial. the new layer is describable - i guess - by a greater number of representations for the transformation group of this pattern. (not sure this is correct though)

more notes for a gauge invariance narrative

[ note: at present this note concatenates these two posts, replacing the first:
more-notes-for-narrative,
ideas-for-invariance-narrative;
which isolate notes from quatum_mechanics and gauge_theory - ]

The scale of the sarh, the building that is modern physical formalism is staggering. I almost cried (i wish i did). if I had been able to better understand I may have shed tears. The quantum field theory formalism is a vast structure of theoretic thought with the complexity of a very large engineering
feat (though it is a description of nature, what appears like a singular engineering feat, except that the term "engineering" would not correctly apply in this case).

It is fine-detailed , and yet one by one the steps in developing / deriving the formulations are simple and elegant but also speak of mathematical genius , command and audacity , although truth be told, proofs are rare in qft.

The indirection in terms of higher-level abstractions of lower-level primitives is manifold.

It is probably the most indirection i've had to deal with wrt any given subject ?

We (rather they, the specialists) are talking about structures of mappings or homomorphisms on groups of continuous unitary transformations (or xform space)
that (also are invariant with respect to their norm as they are unitary ? and ) leave invariant groups of scalar observables (measurable variables, dependent on particles and or time) which are derivable from (in a rather two-way fashion) differential field equations.

In terms of symmetry descriptions of nature are legion and eloquent , thus deserving their own note.

the quantized schrodinger wave equation in the Hamiltonian and state function

has complex solutions. These form a unitary complex vector space with a (complete?) metric (think distance, norm , seminorm, gauge (?)) that has unitary linear self-adjoint operators

much like there are dual spaces in the case of a linear space's inner product
(also think the dual orthogonal spaces of the physical em field ) .

... and so on ...

Though the narrative is neither dense nor tight nor clear yet,
there's enough coalescence to hold one's hat on for the next bout of reading.

All this with a recording of Ramsey Lewis' cover of "That's the way of the world" playing.
This along with a momentous playlist that's been very smooth. Then again, which of my playlists have not been smooth? Well, it did arise that tracks that were dissonant with playlist's general mood push a number of times me to wonder "how did that get on here?" , particularly when trying out the playlists to entertain others. But insofar as a study score, this list is doing fine.
A listing of some of the tracks can be seen in the sidebar on the right .

• first clues on invariance (how or why did invariance and group formalisms take such a central position in qft discourse?)
  
  
  
  
  • but not on gauge or other problems like decoherence.
    
    
  
  
  
  

• in the current formalism (in the language of group theories (and what else?)) , quantities and formula (equations) are looked at
  
  

from the perspective of invariance.

• This seems to have arisen from a categorical treatment of linear transformations to determine or work correctly with the "cannonical components" (p1,…,pn,q1,…,q2) in the 2N phase space of the Hamiltonian formalism.
  
  
  

• This treatment of linear transformations is such as (some kind of ) of operator analysis
  
  
  
  
  • that involves things like commutators (cf. Shankar's commutators discussion in linear vector spaces, ch. 1)
    
    
  
  
  • Poisson brackets, very similar to commutators , as in they have each an exactly similar set of three identities or properties (defining what looks like the same algebra).
    
    
    
    
    • related notes: the cross product , like the dot product depends on the metric of the space , unlike dp it also depends on the handedness of the space or coord system. note also the antisymmetry in axb=-bxa reminiscent of h/skew-symmetry of inner products.
      
      
      
    
    
    
    
  
  
  • constants of motion - these must be like the eigenfunctions of the functions (dependent variables, xforms) in the commutators or Poisson brackets?
    
    
    
    
    • discussion of constants of motion in [cohen]
      
      
    
    
    
    
  
  
  
  

• A result in the Poisson bracket treatment is
  
  
  
  
  • since operators are linear xforms and in turn just functions , variables are considered that are dependent on the canonical coords. , ie,
    
    
    
    
    • f(q,p) , g(p,q)
      
      
    
    
    • note these are note necessarily explicitly dependent on time, though some treatments use f(q,p,t).
      
      
    
    
    
    
    
  
  
  • Without explicitly defining it here, the way the poisson bracket is defined
    
    
    
    sets it equal to the time derivative (rate of change wrt time) of one of the variables (or measurables)
    
    
  
  
  • Thus if the Poisson bracket of a given variable "vanishes" it means the variable remains constant.
    
    
    
    
    
    • because variables (aka functions, lin. xforms, operators, measurables, observables) that are constant have zero-valued derivatives)
      
      
    
    
    • df/dt = 0 ⇒ f = k , k constant.
      
      
    
    
    
    
  
  
  • Thus a zero valued Poisson bracket represents an invariance of symmetry of the transformation or variable.
    
    
  
  
  
  
  

• This is the first reasoned trek from talking about linear transformations and how they affect computed measurables to talking about a symmetry or invariance.
  
  

• Indeed in Fields Siegel opens his discussion of symmetry with commutators and brackets.
  
  
  
  
  
  • He writes:
    
    
  
  
  
  

<pre> "In the Hamiltonian approach to mechanics, both symmetries and dynamics can be expressed conveniently in terms of a \bracket": the Poisson bracket for classical mechanics, the commutator for quantum mechanics. In this formulation, the fundamental variables (operators) are some set of coordinates and their canonically conjugate momenta, as functions of time. The (Heisenberg) operator approach to quantum mechanics then is related to classical mechanics by identifying the semiclassical limit of the commutator as the Poisson bracket: For any functions A and B of p and q, the quantum mechanical commutator" </pre>

– ([Siegel] ch. 1.A.1 "nonrelativity")

• This still does not motivate with any clarity the adoption of group theoretic techniques; except for two things: a trivial observation and something gleaned from the day's review of several QFT books (added within the past day to the references.phys) so far.
  
  

• The trivial observation:
  
  
  
  
  • one thing i can think of as to why use group techniques is that formally, axiomatically, analytically, categorically, algebraically, groups are a generalization of vector spaces and spaces of linear transformations on those vector spaces. Indeed all vectors spaces (including those of linear xforms) are groups.
    
    
  
  
  
  

• (the) Something gleaned so far: (stands to be corrected big time )
  
  
  
  
  
  • The use of invariance classes (such as commutators , Poisson brackets, Lie brackets, or groups) to study analyse or compute operators (lin. transformations) greatly simplifies the recalculations necessary to account for changes in coordinate frames , which requires a set of spatial transformation applications on the systems being analyzed.
    
    
  
  
  • this abstracts or sublimates the operations required to make correct computations for all possible transformations (mutations in reference frames)
    
    
  
  
  • it does so by considering formalisms with only those quantities that are left unchanged by the transformations. This organizes the transformations into classes for which those observables remain constant. The transformation groups then become the symmetry groups for a given quantum field equation which is also called a field theory, or rather a given field's theory.
    
    
  
  
  
  

• later on in the discourse gleaned from texts skimmed earlier ,
  
  
  
  
  • we see the uses of homomorphisms from group transformation spaces to vector spaces (group representations), or perhaps also,
    
    
  
  
  • spaces of such homomorphisms (representations) defined on things other than fields, namely rings, ie, being not linear spaces, but modules.
    
    
    
    
    
    • (since linear vector spaces are defined only on fields (of complex or real scalars).)
      
      
    
    
    
    
  
  
  
  

• it is worth noting also that
  
  
  
  
  • the bracket is like a delta function (eg a kroenecker delta) cf. [Siegel] p. 4.
    
    
  
  
  • ie, similar to or is a metric, a sort of distance
    
    
  
  
  • ie, a norm
    
    
  
  
  
  • hence the speaking norms and seminorms , and hence gauge - since the gauge is "a seminorm" - cf. gauge theory, Norm_(mathematics),
    
    
  
  
  
  

• Hence when we speak of variables and operators that are invariance in the Poisson Bracket , or for which the Poisson Bracket vanishes, we are speaking of the invariance of bracket for that variable, thus a metric,distance,norm,seminorm or gauge invariance.
  
  
  

• both both previous notes: there's a seed for a discussion on gauge as a seminorm taken from a text on norms at wp. (op. cit.)
  
  

ideas for an invariance narrative

• first clues on invariance (how or why did invariance and group formalisms take such a central position in qft discourse?)
  
  
  
  
  • but not on gauge or other problems like decoherence.
    
    
  
  
  
  

• in the current formalism (in the language of group theories (and what else?)) , quantities and formula (equations) are looked at
  
  

from the perspective of invariance.

• This seems to have arisen from a categorical treatment of linear transformations to determine or work correctly with the "cannonical components" (p1,…,pn,q1,…,q2) in the 2N phase space of the Hamiltonian formalism.
  
  
  

• This treatment of linear transformations is such as (some kind of ) of operator analysis
  
  
  
  
  • that involves things like commutators (cf. Shankar's commutators discussion in linear vector spaces, ch. 1)
    
    
  
  
  • Poisson brackets, very similar to commutators , as in they have each an exactly similar set of three identities or properties (defining what looks like the same algebra).
    
    
    
    
    • related notes: the cross product , like the dot product depends on the metric of the space , unlike dp it also depends on the handedness of the space or coord system. note also the antisymmetry in axb=-bxa reminiscent of h/skew-symmetry of inner products.
      
      
      
    
    
    
    
  
  
  • constants of motion - these must be like the eigenfunctions of the functions (dependent variables, xforms) in the commutators or Poisson brackets?
    
    
    
    
    • discussion of constants of motion in [cohen]
      
      
    
    
    
    
  
  
  
  

• A result in the Poisson bracket treatment is
  
  
  
  
  • since operators are linear xforms and in turn just functions , variables are considered that are dependent on the canonical coords. , ie,
    
    
    
    
    • f(q,p) , g(p,q)
      
      
    
    
    • note these are note necessarily explicitly dependent on time, though some treatments use f(q,p,t).
      
      
    
    
    
    
    
  
  
  • Without explicitly defining it here, the way the poisson bracket is defined
    
    
    
    sets it equal to the time derivative (rate of change wrt time) of one of the variables (or measurables)
    
    
  
  
  • Thus if the Poisson bracket of a given variable "vanishes" it means the variable remains constant.
    
    
    
    
    
    • because variables (aka functions, lin. xforms, operators, measurables, observables) that are constant have zero-valued derivatives)
      
      
    
    
    • df/dt = 0 ⇒ f = k , k constant.
      
      
    
    
    
    
  
  
  • Thus a zero valued Poisson bracket represents an invariance of symmetry of the transformation or variable.
    
    
  
  
  
  
  

• This is the first reasoned trek from talking about linear transformations and how they affect computed measurables to talking about a symmetry or invariance.
  
  

• Indeed in Fields Siegel opens his discussion of symmetry with commutators and brackets.
  
  
  
  
  
  • He writes:
    
    
  
  
  
  

<pre> "In the Hamiltonian approach to mechanics, both symmetries and dynamics can be expressed conveniently in terms of a \bracket": the Poisson bracket for classical mechanics, the commutator for quantum mechanics. In this formulation, the fundamental variables (operators) are some set of coordinates and their canonically conjugate momenta, as functions of time. The (Heisenberg) operator approach to quantum mechanics then is related to classical mechanics by identifying the semiclassical limit of the commutator as the Poisson bracket: For any functions A and B of p and q, the quantum mechanical commutator" </pre>

– ([Siegel] ch. 1.A.1 "nonrelativity")

• This still does not motivate with any clarity the adoption of group theoretic techniques; except for two things: a trivial observation and something gleaned from the day's review of several QFT books (added within the past day to the references.phys) so far.
  
  

• The trivial observation:
  
  
  
  
  • one thing i can think of as to why use group techniques is that formally, axiomatically, analytically, categorically, algebraically, groups are a generalization of vector spaces and spaces of linear transformations on those vector spaces. Indeed all vectors spaces (including those of linear xforms) are groups.
    
    
  
  
  
  

• (the) Something gleaned so far: (stands to be corrected big time nohokuku mazalekya )
  
  
  
  
  
  • The use of invariance classes (such as commutators , Poisson brackets, Lie brackets, or groups) to study analyse or compute operators (lin. transformations) greatly simplifies the recalculations necessary to account for changes in coordinate frames , which requires a set of spatial transformation applications on the systems being analyzed.
    
    
  
  
  • this abstracts or sublimates the operations required to make correct computations for all possible transformations (mutations in reference frames)
    
    
  
  
  • it does so by considering formalisms with only those quantities that are left unchanged by the transformations. This organizes the transformations into classes for which those observables remain constant. The transformation groups then become the symmetry groups for a given quantum field equation which is also called a field theory, or rather a given field's theory.
    
    
  
  
  
  

• later on in the discourse gleaned from texts skimmed earlier ,
  
  
  
  
  • we see the uses of homomorphisms from group transformation spaces to vector spaces (group representations), or perhaps also,
    
    
  
  
  • spaces of such homomorphisms (representations) defined on things other than fields, namely rings, ie, being not linear spaces, but modules.
    
    
    
    
    
    • (since linear vector spaces are defined only on fields (of complex or real scalars).)
      
      
    
    
    
    
  
  
  
  

• it is worth noting also that
  
  
  
  
  • the bracket is like a delta function (eg a kroenecker delta) cf. [Siegel] p. 4.
    
    
  
  
  • ie, similar to or is a metric, a sort of distance
    
    
  
  
  • ie, a norm
    
    
  
  
  
  • hence the speaking norms and seminorms , and hence gauge - since the gauge is "a seminorm" - cf. gauge theory, Norm_(mathematics),
    
    
  
  
  
  

• Hence when we speak of variables and operators that are invariance in the Poisson Bracket , or for which the Poisson Bracket vanishes, we are speaking of the invariance of bracket for that variable, thus a metric,distance,norm,seminorm or gauge invariance.
  
  
  

• both both previous notes: there's a seed for a discussion on gauge as a seminorm taken from a text on norms at wp. (op. cit.)
  
  

linear combos everywhere

• a linear combination is a finite sum of the form
  
  
  
  
  • [Shankar] calls it a superposition. ([Shankar] §1.2 p. 9).
    
    
    
    
    • there is also something called the superposition principle:
      
      
      
      
      
      • F(x)=A,F(y)=B ⇒ F(x+y)=A+B
        
        
      
      
      • additivity property of functions making them linear functions,aka lin. operators and lin. maps.
        
        
      
      
      
      
    
    
    
    
  
  
  • a finite sum of that form
    
    
    ⇔ summation of terms
    
    ⇔ sum of products
    
    
  
  
  
  

• instances of linear combination:
  
  
  
  
  
  • linear combinations are thus a key primitive in - mathematically speaking - "nature".
    
    
  
  
  • instances of linear combination in inner products or related by the inner product:
    
    
    
    
    • inner product is a class of operations having the same set of properties,e.g.,
      
      
    
    
    
    • vector dot product
      
      
      
      
      • of two vectors A⋅B=Σaibi , and in its variant, matrix multiplication , thus
        
        
      
      
      
      
    
    
    • matrix multiplication
      
      
      
      
      • , for each row in the mxn premultiplier elements from each column in the row are linearly combined (linearly combine) with elements from the corresponding row element in the postmultiplier giving a single element in the product matrix
        
        
        
      
      
      • cij = ai1b1j + ai2b2j + ai3b3j + … + ainbnj
        
        
      
      
      
      
    
    
    • integration operator
      
      
      
      
      • 1.bis2- in the approximation of integration by finite sums
        
        
      
      
      
      • Σf(ci)Δx , where ci is a point in the ith subinterval
        
        
      
      
      • the integral itself expressing a wide class of functions eg, area,distance,path length,volume,work,flow,etc. = ∫f=limΣf(x)⋅(Δx→0)
        
        
      
      
      • n.b. the integrat operator is a linear transformation, which have linear combination as a property, see below.
        
        
      
      
      
      
    
    
    • N.B. N.B. the equivalence of integrals and the dot product. Explicit examples:
      
      
      
      
      
      • [Waerden]§1: where to express that state functions Ψ are Lebesgue-integrable, he writes <Ψ,Ψ> = ∫ΨΨ*dq, where I take the LHS to be an inner product.
        
        
      
      
      • in investigating, we can take this parallelism further by emphasizing the coef:
        
        
        
        
        • case: dot product, matrix multiplication and determinant: coefs is a "vector" component
          
          
          
        
        
        • case: finite sum: coef. is subinterval width, Δx=b-a/n and kth x is x_k=x0+k(b-a/n)
          
          
        
        
        
        
      
      
      
      
    
    
    • The Unit form , Hermitean product <u,v>
      
      
      
      
      
      • the unit form <v,v> = ∑ c_k^* c_k ([Waerden] §9)
        
        
      
      
      • is pretty damn close to the way
        
        
        
        
        • state functions Ψn of the Schrodinger equation are "integrable in the sense of lebesgue" , ie <Ψ,Ψ>=∫Ψ^*Ψ ([Waerden] §1)
          
          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  • Determinant calculation , for orders 2 and 3 at least ?
    
    
  
  
  • the definition of a vector itself. As a vector is expressible as a linear combination of the magnitudes of its components and the orthonormal basis vectors, better known to us as unit coordinate vectors.
    
    
    
    
    • eg, a vector in Euclidian space may be expressed as
      
      
      V = v_x(1,0,0)+v_y(0,1,0)+v_z(0,0,1) = +v_x.i++v_y.j+v_z.k .
      
      see theorem 12.6 in [Aposotl I].
      
      
    
    
    • a vector in vector space of n-tuples V_n is expressed as a linear combination of the space's unit coordinate vectors (eg, i,j,k for V₃).
      
      
      
      
      
      • [ApostolI]ch15,§15.6: definition: the set of all fin. linear comb. of elmts of linear space S
        
        
        
        
        • satisfies closure,
          
          
        
        
        • is a subspc of S and
          
          
        
        
        • is called the span of S, or subspc spanned by S.
          
          
          
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  • Linear transformation: application of Linear transformation A to finite-dimensional vector space V:
    
    
    
    
    • lin. xforms are matrix multiplications, so by extension of this and by definition ipso facto are also linear combinations
      
      
    
    
    • 
      
      
      
    
    
    • !@ see third property of linear transformations in [Apostol II] 2.1. !@
      
      
    
    
    
    
  
  
  • Gram-schmidt process of orthogonalization
    
    
    
    
    • (mapping among either basis sets or axes or both)
      
      
      
    
    
    • [Apostol II] pp.24-26.
      
      
    
    
    
    
  
  
  
  

• The definition of a linear manifold [von Neumann] §II.1 p. 38.




• "a subset U of a linear space R is called a linear manifold if it contains all linear combinations
  
  a₁f₁ + … + a_kf_k for any k of elements f₁, ... , f_k.




• "[sufficiently requiring]" ( f,g ∈ R ) ⇒ ( af, f+g ∈ R )




• .'. f₁,…,f_k ∈ U ⇒ a₁f₁ , a₁f₁+a₂f₂, a₁f₁+ a₂f₂+a₃f₃, ..., a₁f₁+...+a_kf_k ∈ U


• or, put slightly differently,


• .'. ∀ f₁,...,f_k ∈ U   a₁f₁ , a₁f₁+ a₂f₂, a₁f₁+ a₂f₂+a₃f₃, …, a₁f₁+...+a_kf_k ∈ U.






• U is ⊆ R ⇒ the set a₁f₁ + … + a_kf_k ∀ k=1,2,…, a₁,…,a_k in ℂ, f₁, …, f_k in U , it is a subset of every other linear manifold , which is then said to be spanned by U.

outline

 



 special topics (combinging [Apostol] with others):

 linearity (linear combination linearity and linear independence part 1)

 linearity property

 what does the word linear mean /putting linear in linear algebra


 applications

 linear separability (…) TBD



 linear combination (linear combination linearity and linear independence part 2)


 occurences of linear combination



 linear independence (linear combination linearity and linear independence part 3)



 topics not in [Apostol]


 group theory

 projectors

 closure operators

 tensors - or maybe placed after vectors


 linear algebra glossary






listing of linear algebra topics found in [Apostol]: 


 vectors: vector algebra and vector spaces


 linear spaces = abstract vector spaces

 span

 basis,

 dimension, 


 orthogonalization theorem [Waerden] §9 ; Orthogonality [Apostol I] Ch.12 or 15

 norm, orthonormality , coordinate system

 what about the polar coord. system?






 transformations, linear transformations, linear maps, T:V→V, T:V→W , (matrix algebra|matrices)

 transformations (linear only or in general?) 


 linear transformations are matrices

 examples of linear transformations

 matrices ⇔ linear transformations , 


 why matrices can express transformations

 matrix algebra: typology, multiplication,rank,cofactors,minor,



 reversible transformations = nonsingular transformations




 Determinants

 determinant theorems, properties



 singularity reversibility and determinants

 eigenpolynomials, eigenfunctions and eigenvalues


 characteristic functions and eigenvalues, characteristic equation



 systems of linear equations



 general note on linear algebra


 pedagogy note: linear algebra prerequisite for and segues to abstract algebras like group theory general topology (metric spaces), and i guess manifold theory.

 Applications of linear algebra

 the linear algebra formalism is used extensively in quantum mech and quantum field theory (by way of eigenfunctions, complex linear space, and group theory (eg, symmetry or gauge groups)).

 Just like many natural relationships are observed as (or describable by) differential equations on one hand, or by complex analysis expressions, 


 there are many relationships that are expressable instead only by linear equations (or inequalities, as in the case of linear programming or optimization with constraints)

 A system described by a set of linear equations may be solved 1) using linear algebraic techniques , such as matrix representation and techniques , or what can be called matrix analysis.

 examples of such linear relationships

 occur in macro- and microeconomics, resource planning, expenditure planning, etc. (eg, linear programming problems)




 segue to constraint programming, simplex analysis, linear programming, integer programming, optimization problems , &c.







1) 
solved means finding the unknown function