Merge remote-tracking branch 'origin/main'

notes/2024/math-4a/week-2/main.typ

@@ -1,7 +1,7 @@
 #import "@preview/unequivocal-ams:0.1.1": ams-article, theorem, proof
 
 #show: ams-article.with(
-  title: [Week 2],
+  title: [A Digression on Abstract Linear Algebra],
   authors: (
     (
       name: "Youwen Wu",
@@ -10,6 +10,156 @@
       url: "https://youwen.dev",
     ),
   ),
+  bibliography: bibliography("refs.bib"),
 )
 
-$ integral^oo_(-oo) e^(-x^2) dif x = sqrt(pi) $
+= 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 $(A_(j,k))^T = A_(k,j)$.

notes/2024/math-4a/week-2/refs.bib

@@ -0,0 +1,6 @@
+@misc{chen2024digression,
+  author       = {Evan Chen},
+  title        = {Digression on Type Safety},
+  year         = {2024},
+  howpublished = {\url{https://web.evanchen.cc/upload/1802/tsafe-1802.pdf}},
+}