Compare commits
No commits in common. "dad12b76a49126551d011611bd40e6eb4788897f" and "e596fe72d807a3db1014de2a924aa9ff89d04faf" have entirely different histories.
dad12b76a4
...
e596fe72d8
30 changed files with 0 additions and 486 deletions
|
|
@ -1,35 +0,0 @@
|
||||||
{
|
|
||||||
pkgs,
|
|
||||||
typstPackagesCache,
|
|
||||||
typixLib,
|
|
||||||
cleanTypstSource,
|
|
||||||
...
|
|
||||||
}:
|
|
||||||
let
|
|
||||||
src = cleanTypstSource ./.;
|
|
||||||
commonArgs = {
|
|
||||||
typstSource = "main.typ";
|
|
||||||
|
|
||||||
fontPaths = [
|
|
||||||
# Add paths to fonts here
|
|
||||||
# "${pkgs.roboto}/share/fonts/truetype"
|
|
||||||
];
|
|
||||||
|
|
||||||
virtualPaths = [
|
|
||||||
# Add paths that must be locally accessible to typst here
|
|
||||||
# {
|
|
||||||
# dest = "icons";
|
|
||||||
# src = "${inputs.font-awesome}/svgs/regular";
|
|
||||||
# }
|
|
||||||
];
|
|
||||||
|
|
||||||
XDG_CACHE_HOME = typstPackagesCache;
|
|
||||||
};
|
|
||||||
|
|
||||||
in
|
|
||||||
typixLib.buildTypstProject (
|
|
||||||
commonArgs
|
|
||||||
// {
|
|
||||||
inherit src;
|
|
||||||
}
|
|
||||||
)
|
|
||||||
|
|
@ -1,35 +0,0 @@
|
||||||
{
|
|
||||||
pkgs,
|
|
||||||
typstPackagesCache,
|
|
||||||
typixLib,
|
|
||||||
cleanTypstSource,
|
|
||||||
...
|
|
||||||
}:
|
|
||||||
let
|
|
||||||
src = cleanTypstSource ./.;
|
|
||||||
commonArgs = {
|
|
||||||
typstSource = "main.typ";
|
|
||||||
|
|
||||||
fontPaths = [
|
|
||||||
# Add paths to fonts here
|
|
||||||
# "${pkgs.roboto}/share/fonts/truetype"
|
|
||||||
];
|
|
||||||
|
|
||||||
virtualPaths = [
|
|
||||||
# Add paths that must be locally accessible to typst here
|
|
||||||
# {
|
|
||||||
# dest = "icons";
|
|
||||||
# src = "${inputs.font-awesome}/svgs/regular";
|
|
||||||
# }
|
|
||||||
];
|
|
||||||
|
|
||||||
XDG_CACHE_HOME = typstPackagesCache;
|
|
||||||
};
|
|
||||||
|
|
||||||
in
|
|
||||||
typixLib.buildTypstProject (
|
|
||||||
commonArgs
|
|
||||||
// {
|
|
||||||
inherit src;
|
|
||||||
}
|
|
||||||
)
|
|
||||||
|
|
@ -1,99 +0,0 @@
|
||||||
{
|
|
||||||
"nodes": {
|
|
||||||
"flake-utils": {
|
|
||||||
"inputs": {
|
|
||||||
"systems": "systems"
|
|
||||||
},
|
|
||||||
"locked": {
|
|
||||||
"lastModified": 1726560853,
|
|
||||||
"narHash": "sha256-X6rJYSESBVr3hBoH0WbKE5KvhPU5bloyZ2L4K60/fPQ=",
|
|
||||||
"owner": "numtide",
|
|
||||||
"repo": "flake-utils",
|
|
||||||
"rev": "c1dfcf08411b08f6b8615f7d8971a2bfa81d5e8a",
|
|
||||||
"type": "github"
|
|
||||||
},
|
|
||||||
"original": {
|
|
||||||
"owner": "numtide",
|
|
||||||
"repo": "flake-utils",
|
|
||||||
"type": "github"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"nixpkgs": {
|
|
||||||
"locked": {
|
|
||||||
"lastModified": 1729256560,
|
|
||||||
"narHash": "sha256-/uilDXvCIEs3C9l73JTACm4quuHUsIHcns1c+cHUJwA=",
|
|
||||||
"owner": "NixOS",
|
|
||||||
"repo": "nixpkgs",
|
|
||||||
"rev": "4c2fcb090b1f3e5b47eaa7bd33913b574a11e0a0",
|
|
||||||
"type": "github"
|
|
||||||
},
|
|
||||||
"original": {
|
|
||||||
"owner": "NixOS",
|
|
||||||
"ref": "nixos-unstable",
|
|
||||||
"repo": "nixpkgs",
|
|
||||||
"type": "github"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"root": {
|
|
||||||
"inputs": {
|
|
||||||
"flake-utils": "flake-utils",
|
|
||||||
"nixpkgs": "nixpkgs",
|
|
||||||
"typix": "typix",
|
|
||||||
"typst-packages": "typst-packages"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"systems": {
|
|
||||||
"locked": {
|
|
||||||
"lastModified": 1681028828,
|
|
||||||
"narHash": "sha256-Vy1rq5AaRuLzOxct8nz4T6wlgyUR7zLU309k9mBC768=",
|
|
||||||
"owner": "nix-systems",
|
|
||||||
"repo": "default",
|
|
||||||
"rev": "da67096a3b9bf56a91d16901293e51ba5b49a27e",
|
|
||||||
"type": "github"
|
|
||||||
},
|
|
||||||
"original": {
|
|
||||||
"owner": "nix-systems",
|
|
||||||
"repo": "default",
|
|
||||||
"type": "github"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"typix": {
|
|
||||||
"inputs": {
|
|
||||||
"nixpkgs": [
|
|
||||||
"nixpkgs"
|
|
||||||
]
|
|
||||||
},
|
|
||||||
"locked": {
|
|
||||||
"lastModified": 1728290750,
|
|
||||||
"narHash": "sha256-piLZT8398O69hy0e3gX8hUWqSzbCslMCt10pT8d6c8E=",
|
|
||||||
"owner": "loqusion",
|
|
||||||
"repo": "typix",
|
|
||||||
"rev": "62d032735ad32a9a90225c06cb0857677fc753ee",
|
|
||||||
"type": "github"
|
|
||||||
},
|
|
||||||
"original": {
|
|
||||||
"owner": "loqusion",
|
|
||||||
"repo": "typix",
|
|
||||||
"type": "github"
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"typst-packages": {
|
|
||||||
"flake": false,
|
|
||||||
"locked": {
|
|
||||||
"lastModified": 1729252723,
|
|
||||||
"narHash": "sha256-xdaQYNw1GApaJnXJO4WnUTNuYeRXisxIUSuAzVXFEW4=",
|
|
||||||
"owner": "typst",
|
|
||||||
"repo": "packages",
|
|
||||||
"rev": "9bcf190c85da91be6df9f5fc25a93ce5812489cc",
|
|
||||||
"type": "github"
|
|
||||||
},
|
|
||||||
"original": {
|
|
||||||
"owner": "typst",
|
|
||||||
"repo": "packages",
|
|
||||||
"type": "github"
|
|
||||||
}
|
|
||||||
}
|
|
||||||
},
|
|
||||||
"root": "root",
|
|
||||||
"version": 7
|
|
||||||
}
|
|
||||||
|
|
@ -1,92 +0,0 @@
|
||||||
{
|
|
||||||
description = "Compiling work in 2024 in a reproducible manner";
|
|
||||||
|
|
||||||
inputs = {
|
|
||||||
nixpkgs.url = "github:NixOS/nixpkgs/nixos-unstable";
|
|
||||||
|
|
||||||
typix = {
|
|
||||||
url = "github:loqusion/typix";
|
|
||||||
inputs.nixpkgs.follows = "nixpkgs";
|
|
||||||
};
|
|
||||||
|
|
||||||
flake-utils.url = "github:numtide/flake-utils";
|
|
||||||
|
|
||||||
typst-packages = {
|
|
||||||
url = "github:typst/packages";
|
|
||||||
flake = false;
|
|
||||||
};
|
|
||||||
|
|
||||||
# Example of downloading icons from a non-flake source
|
|
||||||
# font-awesome = {
|
|
||||||
# url = "github:FortAwesome/Font-Awesome";
|
|
||||||
# flake = false;
|
|
||||||
# };
|
|
||||||
};
|
|
||||||
|
|
||||||
outputs =
|
|
||||||
inputs@{
|
|
||||||
nixpkgs,
|
|
||||||
typix,
|
|
||||||
flake-utils,
|
|
||||||
...
|
|
||||||
}:
|
|
||||||
flake-utils.lib.eachDefaultSystem (
|
|
||||||
system:
|
|
||||||
let
|
|
||||||
pkgs = nixpkgs.legacyPackages.${system};
|
|
||||||
in
|
|
||||||
let
|
|
||||||
typstPackagesSrc = "${inputs.typst-packages}/packages";
|
|
||||||
|
|
||||||
typstPackagesCache = pkgs.stdenv.mkDerivation {
|
|
||||||
name = "typst-packages-cache";
|
|
||||||
src = typstPackagesSrc;
|
|
||||||
dontBuild = true;
|
|
||||||
installPhase = ''
|
|
||||||
mkdir -p "$out/typst/packages"
|
|
||||||
cp -LR --reflink=auto --no-preserve=mode -t "$out/typst/packages" "$src"/*
|
|
||||||
'';
|
|
||||||
};
|
|
||||||
typixLib = typix.lib.${system};
|
|
||||||
in
|
|
||||||
let
|
|
||||||
alexandriaLib = import ./nix/lib {
|
|
||||||
inherit pkgs typixLib typstPackagesCache;
|
|
||||||
};
|
|
||||||
|
|
||||||
in
|
|
||||||
{
|
|
||||||
# checks = {
|
|
||||||
# inherit build-drv build-script watch-script;
|
|
||||||
# };
|
|
||||||
|
|
||||||
legacyPackages = {
|
|
||||||
phil-1 = {
|
|
||||||
paper-1 = alexandriaLib.callTypstProject (import ./documents/by-course/phil-1/paper-1/package.nix);
|
|
||||||
};
|
|
||||||
digression-linear-algebra = alexandriaLib.callTypstProject (import ./documents/by-name/digression-linear-algebra/package.nix);
|
|
||||||
};
|
|
||||||
# apps = rec {
|
|
||||||
# default = watch;
|
|
||||||
# build = flake-utils.lib.mkApp {
|
|
||||||
# drv = build-script;
|
|
||||||
# };
|
|
||||||
# watch = flake-utils.lib.mkApp {
|
|
||||||
# drv = watch-script;
|
|
||||||
# };
|
|
||||||
# };
|
|
||||||
#
|
|
||||||
# devShells.default = typixLib.devShell {
|
|
||||||
# inherit (commonArgs) fontPaths virtualPaths;
|
|
||||||
# packages = [
|
|
||||||
# # WARNING: Don't run `typst-build` directly, instead use `nix run .#build`
|
|
||||||
# # See https://github.com/loqusion/typix/issues/2
|
|
||||||
# # build-script
|
|
||||||
# watch-script
|
|
||||||
# # More packages can be added here, like typstfmt
|
|
||||||
# pkgs.typstyle
|
|
||||||
# ];
|
|
||||||
# };
|
|
||||||
}
|
|
||||||
);
|
|
||||||
}
|
|
||||||
|
|
@ -1,16 +0,0 @@
|
||||||
{
|
|
||||||
pkgs,
|
|
||||||
typixLib,
|
|
||||||
typstPackagesCache,
|
|
||||||
cleanTypstSource,
|
|
||||||
...
|
|
||||||
}:
|
|
||||||
package:
|
|
||||||
package {
|
|
||||||
inherit
|
|
||||||
pkgs
|
|
||||||
typixLib
|
|
||||||
typstPackagesCache
|
|
||||||
cleanTypstSource
|
|
||||||
;
|
|
||||||
}
|
|
||||||
|
|
@ -1,18 +0,0 @@
|
||||||
{ pkgs, ... }:
|
|
||||||
src:
|
|
||||||
let
|
|
||||||
inherit (pkgs) lib;
|
|
||||||
in
|
|
||||||
lib.cleanSourceWith {
|
|
||||||
src = lib.cleanSource src;
|
|
||||||
filter =
|
|
||||||
path: type:
|
|
||||||
let
|
|
||||||
isTypstSource = lib.hasSuffix ".typ" path || lib.hasSuffix ".bib" path;
|
|
||||||
isImage = lib.hasSuffix ".png" path || lib.hasSuffix ".jpeg" path || lib.hasSuffix ".jpg" path;
|
|
||||||
isSpecialFile = builtins.elem (builtins.baseNameOf path) [
|
|
||||||
"typst.toml"
|
|
||||||
];
|
|
||||||
in
|
|
||||||
type == "directory" || isTypstSource || isSpecialFile || isImage;
|
|
||||||
}
|
|
||||||
|
|
@ -1,19 +0,0 @@
|
||||||
{
|
|
||||||
pkgs,
|
|
||||||
typstPackagesCache,
|
|
||||||
typixLib,
|
|
||||||
}:
|
|
||||||
let
|
|
||||||
defaultArgs = {
|
|
||||||
inherit pkgs typstPackagesCache typixLib;
|
|
||||||
};
|
|
||||||
in
|
|
||||||
rec {
|
|
||||||
cleanTypstSource = (import ./cleanTypstSource.nix) defaultArgs;
|
|
||||||
callTypstProject = (import ./callTypstProject.nix) (
|
|
||||||
defaultArgs
|
|
||||||
// {
|
|
||||||
inherit cleanTypstSource;
|
|
||||||
}
|
|
||||||
);
|
|
||||||
}
|
|
||||||
|
|
@ -1,165 +0,0 @@
|
||||||
#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof
|
|
||||||
|
|
||||||
#show: ams-article.with(
|
|
||||||
title: [A Digression on Abstract Linear Algebra],
|
|
||||||
authors: (
|
|
||||||
(
|
|
||||||
name: "Youwen Wu",
|
|
||||||
organization: [University of California, Santa Barbara],
|
|
||||||
email: "youwen@ucsb.edu",
|
|
||||||
url: "https://youwen.dev",
|
|
||||||
),
|
|
||||||
),
|
|
||||||
bibliography: bibliography("refs.bib"),
|
|
||||||
)
|
|
||||||
|
|
||||||
= Introduction
|
|
||||||
|
|
||||||
Many introductory linear algebra classes focus on _application_. In general,
|
|
||||||
this is a red herring and is engineer-speak for "we will teach you how to
|
|
||||||
crunch numbers with no regard for conceptual understanding."
|
|
||||||
|
|
||||||
If you are a math major (or math-adjacent, such as Computer Science), this
|
|
||||||
class is essentially useless for you. You will learn how to perform trivial
|
|
||||||
numerical operations such as the _matrix multiplication_, _matrix-vector
|
|
||||||
multiplication_, _row reduction_, and other trite tasks better suited for
|
|
||||||
computers.
|
|
||||||
|
|
||||||
If you are taking this course, you might as well learn linear algebra properly.
|
|
||||||
Otherwise, you will have to re-learn it later on, anyways. Completing a math
|
|
||||||
course without gaining a theoretical appreciation for the topics at hand is an
|
|
||||||
unequivocal waste of time. I have prepared this brief crash course designed to
|
|
||||||
fill in the theoretical gaps left by this class.
|
|
||||||
|
|
||||||
= Basic Notions
|
|
||||||
|
|
||||||
== Vector spaces
|
|
||||||
|
|
||||||
Before we can understand vectors, we need to first discuss _vector spaces_. Thus
|
|
||||||
far, you have likely encountered vectors primarily in physics classes,
|
|
||||||
generally in the two-dimensional plane. You may conceptualize them as arrows in
|
|
||||||
space. For vectors of size $>3$, a hand waving argument is made that they are
|
|
||||||
essentially just arrows in higher dimensional spaces.
|
|
||||||
|
|
||||||
It is helpful to take a step back from this primitive geometric understanding
|
|
||||||
of the vector. Let us build up a rigorous idea of vectors from first
|
|
||||||
principles.
|
|
||||||
|
|
||||||
=== Vector axioms
|
|
||||||
|
|
||||||
The so-called _axioms_ of a _vector space_ (which we'll call the vector space
|
|
||||||
$V$) are as follows:
|
|
||||||
|
|
||||||
#enum[
|
|
||||||
Commutativity: $u + v = v + u, " " forall u,v in V$
|
|
||||||
][
|
|
||||||
Associativity: $(u + v) + w = u + (v + w), " " forall u,v,w in V$
|
|
||||||
][
|
|
||||||
Zero vector: $exists$ a special vector, denoted $0$, such that $v + 0 = v, " " forall v in V$
|
|
||||||
][
|
|
||||||
Additive inverse: $forall v in V, " " exists w in V "such that" v + w = 0$. Such an additive inverse is generally denoted $-v$
|
|
||||||
][
|
|
||||||
Multiplicative identity: $1 v = v, " " forall v in V$
|
|
||||||
][
|
|
||||||
Multiplicative associativity: $(alpha beta) v = alpha (beta v) " " forall v in V, "scalars" alpha, beta$
|
|
||||||
][
|
|
||||||
Distributive property for vectors: $alpha (u + v) = alpha u + alpha v " " forall u,v in V, "scalars" alpha$
|
|
||||||
][
|
|
||||||
Distributive property for scalars: $(alpha + beta) v = alpha v + beta v " " forall v in V, " scalars" alpha, beta$
|
|
||||||
]
|
|
||||||
|
|
||||||
It is easy to show that the zero vector $0$ and the additive inverse $-v$ are
|
|
||||||
_unique_. We leave the proof of this fact as an exercise.
|
|
||||||
|
|
||||||
These may seem difficult to memorize, but they are essentially the same
|
|
||||||
familiar algebraic properties of numbers you know from high school. The
|
|
||||||
important thing to remember is which operations are valid for what objects. For
|
|
||||||
example, you cannot add a vector and scalar, as it does not make sense.
|
|
||||||
|
|
||||||
_Remark_. For those of you versed in computer science, you may recognize this
|
|
||||||
as essentially saying that you must ensure your operations are _type-safe_.
|
|
||||||
Adding a vector and scalar is not just false, it is an _invalid question_
|
|
||||||
entirely because vectors and scalars and different types of mathematical
|
|
||||||
objects. See #cite(<chen2024digression>, form: "prose") for more.
|
|
||||||
|
|
||||||
=== Vectors big and small
|
|
||||||
|
|
||||||
In order to begin your descent into what mathematicians colloquially recognize
|
|
||||||
as _abstract vapid nonsense_, let's discuss which fields constitute a vector space. We
|
|
||||||
have the familiar space where all scalars are real numbers, or $RR$. We
|
|
||||||
generally discuss 2-D or 3-D vectors, corresponding to vectors of length 2 or
|
|
||||||
3; in our case, $RR^2$ and $RR^3$.
|
|
||||||
|
|
||||||
However, vectors in $RR$ can really be of any length. Discard your primitive
|
|
||||||
conception of vectors as arrows in space. Vectors are simply arbitrary length
|
|
||||||
lists of numbers (for the computer science folk: think C++ `std::vector`).
|
|
||||||
|
|
||||||
_Example_. $ vec(1,2,3,4,5,6,7,8,9) $
|
|
||||||
|
|
||||||
Moreover, vectors need not be in $RR$ at all. Recall that a vector space need
|
|
||||||
only satisfy the aforementioned _axioms of a vector space_.
|
|
||||||
|
|
||||||
_Example_. The vector space $CC$ is similar to $RR$, except it includes complex
|
|
||||||
numbers. All complex vector spaces are real vector spaces (as you can simply
|
|
||||||
restrict them to only use the real numbers), but not the other way around.
|
|
||||||
|
|
||||||
In general, we can have a vector space where the scalars are in an arbitrary
|
|
||||||
field $FF$, as long as the axioms are satisfied.
|
|
||||||
|
|
||||||
_Example_. The vector space of all polynomials of degree 3, or $PP^3$. It is
|
|
||||||
not yet clear what this vector may look like. We shall return to this example
|
|
||||||
once we discuss _basis_.
|
|
||||||
|
|
||||||
== Vector addition. Multiplication
|
|
||||||
|
|
||||||
Vector addition, represented by $+$, and multiplication, represented by the
|
|
||||||
$dot$ (dot) operator, can be done entrywise.
|
|
||||||
|
|
||||||
_Example._
|
|
||||||
|
|
||||||
$
|
|
||||||
vec(1,2,3) + vec(4,5,6) = vec(1 + 4, 2 + 5, 3 + 6) = vec(5,7,9)
|
|
||||||
$
|
|
||||||
$
|
|
||||||
vec(1,2,3) dot vec(4,5,6) = vec(1 dot 4, 2 dot 5, 3 dot 6) = vec(4,10,18)
|
|
||||||
$
|
|
||||||
|
|
||||||
This is simple enough to understand. Again, the difficulty is simply ensuring
|
|
||||||
that you always perform operations with the correct _types_. For example, once
|
|
||||||
we introduce matrices, it doesn't make sense to multiply or add vectors and
|
|
||||||
matrices in this fashion.
|
|
||||||
|
|
||||||
== Vector-scalar multiplication
|
|
||||||
|
|
||||||
Multiplying a vector by a scalar simply results in each entry of the vector
|
|
||||||
being multiplied by the scalar.
|
|
||||||
|
|
||||||
_Example_.
|
|
||||||
|
|
||||||
$ beta vec(a, b, c) = vec(beta dot a, beta dot b, beta dot c) $
|
|
||||||
|
|
||||||
== Matrices
|
|
||||||
|
|
||||||
Before discussing any properties of matrices, let's simply reiterate what we
|
|
||||||
learned in class about their notation. We say a matrix with rows of length $m$,
|
|
||||||
and columns of size $n$ (in less precise terms, a matrix with length $m$ and
|
|
||||||
height $n$) is a $m times n$ matrix.
|
|
||||||
|
|
||||||
Given a matrix
|
|
||||||
|
|
||||||
$ A = mat(1,2,3;4,5,6;7,8,9) $
|
|
||||||
|
|
||||||
we refer to the entry in row $j$ and column $k$ as $A_(j,k)$ .
|
|
||||||
|
|
||||||
=== Matrix transpose
|
|
||||||
|
|
||||||
A formalism that is useful later on is called the _transpose_, and we obtain it
|
|
||||||
from a matrix $A$ by switching all the rows and columns. More precisely, each
|
|
||||||
row becomes a column instead. We use the notation $A^T$ to represent the
|
|
||||||
transpose of $A$.
|
|
||||||
|
|
||||||
$
|
|
||||||
mat(1,2,3;4,5,6)^T = mat(1,4;2,5;3,6)
|
|
||||||
$
|
|
||||||
|
|
||||||
Formally, we can say $(A_(j,k))^T = A_(k,j)$.
|
|
||||||
|
|
@ -1,6 +0,0 @@
|
||||||
@misc{chen2024digression,
|
|
||||||
author = {Evan Chen},
|
|
||||||
title = {Digression on Type Safety},
|
|
||||||
year = {2024},
|
|
||||||
howpublished = {\url{https://web.evanchen.cc/upload/1802/tsafe-1802.pdf}},
|
|
||||||
}
|
|
||||||
|
|
@ -1 +0,0 @@
|
||||||
/nix/store/fv6y2dqv26lw4gzb2pf6ms5hyikpbkm9-typst
|
|
||||||
0
flake.nix
Normal file
0
flake.nix
Normal file
Loading…
Reference in a new issue