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Since a linear form is completely determined by how it acts on a basis, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the

V

, and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.

Infinite dimension seems more interesting however, see: en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278#Infinite-dimensional_case

One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space

V

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Linear operator by

Ciro Santilli 40 Updated 2025-07-16

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We define it as a linear map where the domain is the same as the image, i.e. an endofunction.

Examples:

a 2x2 matrix can represent a linear map from $R^{2}$ to $R^{2}$ , so which is a linear operator
the derivative is a linear map from $C^{\infty}$ to $C^{\infty}$ , so which is also a linear operator

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Adjoint operator by

Ciro Santilli 40 Updated 2025-07-16

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Given a linear operator

A

over a space

S

that has a inner product defined, we define the adjoint operator

A^{†}

(the

†

symbol is called "dagger") as the unique operator that satisfies:

\forall v, w \in S, < A v, w >=< v, A^{†} w >

(1)

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Self-adjoint operator by

Ciro Santilli 40 Updated 2025-07-16

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Bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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Analogous to a linear form, a bilinear form is a Bilinear map where the image is the underlying field of the vector space, e.g.

R^{n} \times R^{m} \to R

Some definitions require both of the input spaces to be the same, e.g.

R^{n} \times R^{n} \to R

, but it doesn't make much different in general.

The most important example of a bilinear form is the dot product. It is only defined if both the input spaces are the same.

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Effect of a change of basis on the matrix of a bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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C

is the change of basis matrix, then the matrix representation of a bilinear form

M

that looked like:

B (x, y) = x^{T} M y

(1)

then the matrix in the new basis is:

C^{T} MC

(2)

Sylvester's law of inertia then tells us that the number of positive, negative and 0 eigenvalues of both of those matrices is the same.

Proof: the value of a given bilinear form cannot change due to a change of basis, since the bilinear form is just a function, and does not depend on the choice of basis. The only thing that change is the matrix representation of the form. Therefore, we must have:

x^{T} M y = x_{n e w}^{T} M_{n e w} y_{n e w}

(3)

and in the new basis:

x = C x_{n e w} y = C y_{n e w} x_{n e w}^{T} M_{n e w} y_{n e w} = x^{T} M y = (C x_{n e w})^{T} M (C y_{n e w}) = x_{n e w}^{T} (C^{T} MC) y_{n e w}

(4)

and so since:

\forall x_{n e w}, y_{n e w} x_{n e w}^{T} M_{n e w} y_{n e w} = x_{n e w}^{T} (C^{T} MC) y_{n e w} ⟹ M_{n e w} = C^{T} MC

(5)

proofwiki.org/wiki/Matrix_of_Bilinear_Form_Under_Change_of_Basis

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Multilinear form by

Ciro Santilli 40 Updated 2025-07-16

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See form.

Analogous to a linear form, a multilinear form is a Multilinear map where the image is the underlying field of the vector space, e.g.

R^{n_{1}} \times R^{n_{2}} \times R^{n_{2}} \to R

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Symmetric bilinear map by

Ciro Santilli 40 Updated 2025-07-16

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Subcase of symmetric multilinear map:

f (x, y) = f (y, x)

(1)

Requires the two inputs

x

and

y

to be in the same vector space of course.

The most important example is the dot product, which is also a positive definite symmetric bilinear form.

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Symmetric bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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symmetric bilinear maps that is also a bilinear form.

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Hermitian form by

Ciro Santilli 40 Updated 2025-07-16

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The complex number analogue of a symmetric bilinear form.

The prototypical example of it is the complex dot product.

Note that this form is neither strictly symmetric, it satisfies:

< x, y >= \overline{< y, x >}

(1)

where the over bar indicates the complex conjugate, nor is it linear for complex scalar multiplication on the second argument.

Bibliography:

mathworld.wolfram.com/HermitianForm.html

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Oxford physics course handbook by

Ciro Santilli 40 Updated 2025-07-16

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The normal navigation to them was paywalled, but the static files are served without login checks if you know their URL. One way to go about it is to search by prefix on the Wayback Machine: web.archive.org/web/*/https://www2.physics.ox.ac.uk/sites/default/files/contentblock/2011/06/03/*

The last handbooks we can find are 2020/2021, they might have move to a new more properly paywalled location after that year.

2020/2021:
2019/2020. They seem to have split the handbook up per year after some point.
2016/2017: www2.physics.ox.ac.uk/sites/default/files/2011-06-03/course_v3_pdf_80151.pdf 2022 archive: web.archive.org/web/20221229021312/https://www2.physics.ox.ac.uk/sites/default/files/2011-06-03/course_v3_pdf_80151.pdf
This older handbook had a more detailed course breakdown in terms of terms and weeks, e.g. on page 19.

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Matrix representation of a Hermitian form by

Ciro Santilli 40 Updated 2025-07-16

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;

A Hermitian matrix.

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Positive definite symmetric bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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Symmetric bilinear form that is also positive definite, i.e.:

\forall x, B (x, x) > 0

(1)

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Matrix representation of a positive definite symmetric bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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A positive definite matrix that is also a symmetric matrix.

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Representation theory by

Ciro Santilli 40 Updated 2025-07-16

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Basically, a "representation" means associating each group element as an invertible matrices, i.e. a matrix in (possibly some subset of)

G L (n)

, that has the same properties as the group.

Or in other words, associating to the more abstract notion of a group more concrete objects with which we are familiar (e.g. a matrix).

Each such matrix then represents one specific element of the group.

This is basically what everyone does (or should do!) when starting to study Lie groups: we start looking at matrix Lie groups, which are very concrete.

Or more precisely, mapping each group element to a linear map over some vector field

V

(which can be represented by a matrix infinite dimension), in a way that respects the group operations:

R (g) : G \to G L (V)

(1)

As shown at Physics from Symmetry by Jakob Schwichtenberg (2015)

page 51, a representation is not unique, we can even use matrices of different dimensions to represent the same group
3.6 classifies the representations of $S U (2)$ . There is only one possibility per dimension!
3.7 "The Lorentz Group O(1,3)" mentions that even for a "simple" group such as the Lorentz group, not all representations can be described in terms of matrices, and that we can construct such representations with the help of Lie group theory, and that they have fundamental physical application

Motivation:

math.stackexchange.com/questions/1628464/what-is-representation-theory

Bibliography:

www.youtube.com/watch?v=9rDzaKASMTM "RT1: Representation Theory Basics" by MathDoctorBob (2011). Too much theory, give me the motivation!
www.quantamagazine.org/the-useless-perspective-that-transformed-mathematics-20200609 The "Useless" Perspective That Transformed Mathematics by Quanta Magazine (2020). Maybe there is something in there amidst the "the reader might not know what a matrix is" stuff.

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Skew-symmetric bilinear map by

Ciro Santilli 40 Updated 2025-07-16

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Subcase of antisymmetric multilinear map:

f (x, y) = - f (y, x)

(1)

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Skew-symmetric bilinear form by

Ciro Santilli 40 Updated 2025-07-16

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Skew-symmetric bilinear map that is also a bilinear form.

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Tesselation by

Ciro Santilli 40 Updated 2025-07-16

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either https://OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade