165 lines
6.3 KiB
Text
165 lines
6.3 KiB
Text
#import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof
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#show: ams-article.with(
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title: [A Digression on Abstract Linear Algebra],
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authors: (
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(
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name: "Youwen Wu",
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organization: [University of California, Santa Barbara],
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email: "youwen@ucsb.edu",
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url: "https://youwen.dev",
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),
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),
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bibliography: bibliography("refs.bib"),
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)
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= Introduction
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Many introductory linear algebra classes focus on _application_. In general,
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this is a red herring and is engineer-speak for "we will teach you how to
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crunch numbers with no regard for conceptual understanding."
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If you are a math major (or math-adjacent, such as Computer Science), this
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class is essentially useless for you. You will learn how to perform trivial
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numerical operations such as the _matrix multiplication_, _matrix-vector
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multiplication_, _row reduction_, and other trite tasks better suited for
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computers.
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If you are taking this course, you might as well learn linear algebra properly.
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Otherwise, you will have to re-learn it later on, anyways. Completing a math
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course without gaining a theoretical appreciation for the topics at hand is an
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unequivocal waste of time. I have prepared this brief crash course designed to
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fill in the theoretical gaps left by this class.
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= Basic Notions
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== Vector spaces
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Before we can understand vectors, we need to first discuss _vector spaces_. Thus
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far, you have likely encountered vectors primarily in physics classes,
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generally in the two-dimensional plane. You may conceptualize them as arrows in
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space. For vectors of size $>3$, a hand waving argument is made that they are
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essentially just arrows in higher dimensional spaces.
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It is helpful to take a step back from this primitive geometric understanding
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of the vector. Let us build up a rigorous idea of vectors from first
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principles.
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=== Vector axioms
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The so-called _axioms_ of a _vector space_ (which we'll call the vector space
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$V$) are as follows:
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#enum[
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Commutativity: $u + v = v + u, " " forall u,v in V$
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][
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Associativity: $(u + v) + w = u + (v + w), " " forall u,v,w in V$
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][
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Zero vector: $exists$ a special vector, denoted $0$, such that $v + 0 = v, " " forall v in V$
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][
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Additive inverse: $forall v in V, " " exists w in V "such that" v + w = 0$. Such an additive inverse is generally denoted $-v$
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][
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Multiplicative identity: $1 v = v, " " forall v in V$
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][
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Multiplicative associativity: $(alpha beta) v = alpha (beta v) " " forall v in V, "scalars" alpha, beta$
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][
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Distributive property for vectors: $alpha (u + v) = alpha u + alpha v " " forall u,v in V, "scalars" alpha$
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][
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Distributive property for scalars: $(alpha + beta) v = alpha v + beta v " " forall v in V, " scalars" alpha, beta$
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]
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It is easy to show that the zero vector $0$ and the additive inverse $-v$ are
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_unique_. We leave the proof of this fact as an exercise.
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These may seem difficult to memorize, but they are essentially the same
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familiar algebraic properties of numbers you know from high school. The
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important thing to remember is which operations are valid for what objects. For
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example, you cannot add a vector and scalar, as it does not make sense.
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_Remark_. For those of you versed in computer science, you may recognize this
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as essentially saying that you must ensure your operations are _type-safe_.
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Adding a vector and scalar is not just false, it is an _invalid question_
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entirely because vectors and scalars and different types of mathematical
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objects. See #cite(<chen2024digression>, form: "prose") for more.
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=== Vectors big and small
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In order to begin your descent into what mathematicians colloquially recognize
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as _abstract vapid nonsense_, let's discuss which fields constitute a vector space. We
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have the familiar space where all scalars are real numbers, or $RR$. We
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generally discuss 2-D or 3-D vectors, corresponding to vectors of length 2 or
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3; in our case, $RR^2$ and $RR^3$.
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However, vectors in $RR$ can really be of any length. Discard your primitive
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conception of vectors as arrows in space. Vectors are simply arbitrary length
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lists of numbers (for the computer science folk: think C++ `std::vector`).
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_Example_. $ vec(1,2,3,4,5,6,7,8,9) $
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Moreover, vectors need not be in $RR$ at all. Recall that a vector space need
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only satisfy the aforementioned _axioms of a vector space_.
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_Example_. The vector space $CC$ is similar to $RR$, except it includes complex
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numbers. All complex vector spaces are real vector spaces (as you can simply
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restrict them to only use the real numbers), but not the other way around.
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In general, we can have a vector space where the scalars are in an arbitrary
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field $FF$, as long as the axioms are satisfied.
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_Example_. The vector space of all polynomials of degree 3, or $PP^3$. It is
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not yet clear what this vector may look like. We shall return to this example
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once we discuss _basis_.
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== Vector addition. Multiplication
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Vector addition, represented by $+$, and multiplication, represented by the
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$dot$ (dot) operator, can be done entrywise.
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_Example._
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$
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vec(1,2,3) + vec(4,5,6) = vec(1 + 4, 2 + 5, 3 + 6) = vec(5,7,9)
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$
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$
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vec(1,2,3) dot vec(4,5,6) = vec(1 dot 4, 2 dot 5, 3 dot 6) = vec(4,10,18)
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$
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This is simple enough to understand. Again, the difficulty is simply ensuring
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that you always perform operations with the correct _types_. For example, once
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we introduce matrices, it doesn't make sense to multiply or add vectors and
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matrices in this fashion.
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== Vector-scalar multiplication
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Multiplying a vector by a scalar simply results in each entry of the vector
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being multiplied by the scalar.
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_Example_.
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$ beta vec(a, b, c) = vec(beta dot a, beta dot b, beta dot c) $
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== Matrices
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Before discussing any properties of matrices, let's simply reiterate what we
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learned in class about their notation. We say a matrix with rows of length $m$,
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and columns of size $n$ (in less precise terms, a matrix with length $m$ and
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height $n$) is a $m times n$ matrix.
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Given a matrix
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$ A = mat(1,2,3;4,5,6;7,8,9) $
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we refer to the entry in row $j$ and column $k$ as $A_(j,k)$ .
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=== Matrix transpose
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A formalism that is useful later on is called the _transpose_, and we obtain it
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from a matrix $A$ by switching all the rows and columns. More precisely, each
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row becomes a column instead. We use the notation $A^T$ to represent the
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transpose of $A$.
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$
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mat(1,2,3;4,5,6)^T = mat(1,4;2,5;3,6)
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$
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Formally, we can say $(A_(j,k))^T = A_(k,j)$.
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