Compare commits
No commits in common. "6f693b9da50f4e87be299eaee36a0ff690d68744" and "5128e83d2e8fffc48d49eb8ab648e8daca14dcf9" have entirely different histories.
6f693b9da5
...
5128e83d2e
3 changed files with 2 additions and 158 deletions
Binary file not shown.
|
|
@ -1,7 +1,7 @@
|
|||
#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof
|
||||
|
||||
#show: ams-article.with(
|
||||
title: [A Digression on Abstract Linear Algebra],
|
||||
title: [Week 2],
|
||||
authors: (
|
||||
(
|
||||
name: "Youwen Wu",
|
||||
|
|
@ -10,156 +10,6 @@
|
|||
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 $(A_(j,k))^T = A_(k,j)$.
|
||||
$ integral^oo_(-oo) e^(-x^2) dif x = sqrt(pi) $
|
||||
|
|
|
|||
|
|
@ -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}},
|
||||
}
|
||||
Loading…
Reference in a new issue