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zzEZ)u%OVV_iSannUlm7BG%UJTS2dpIO*W!2xKYrz#Q28wW8jB2w7&zYQmHHTU<}T? zK3^33oDCF}kqwLSdoSZ%c=D(7Mg{%3E)tm4k5jV}K{A%`ubg2pVD3t{GfG{)k-`zKH8OQEE*Ir70EA)Q4&!PSg@%%~QC@U)SFk~IEO;E;BI{}9(bSNZWAfIjnSeY)aO)Ok)* ztuEZ&=X?&?C2t(=xkwYF> z!TkSNq*6FY0~X8&DGCRPKyJc8B(j|BDD3|)QSfkcb8=Hyq5lix;NxQFXQ%jAErrv6 zV_<$BzW=~@*f}AA5g>kmi|5sl|4IL?@S5e}e;vsGiM-F#8e_?#Q|AB$IIC%d}`CoEi9^O~a_)q%p5#-?Ie06UB3*!TG{^Jt= zhukX$@cpmh=iuYyeszlfOOT6$`#53vG^pA1cg4s|994U2@5k)F#LaKt@Ue_ e;XelD=5Ffh{@K;i3WbA@oA3$, a hand waving argument is made that they are +essentially just arrows in higher dimensional spaces. + +It is helpful to take a step back from this primitive geometric understanding +of the vector. Let us build up a rigorous idea of vectors from first +principles. + +=== Vector axioms + +The so-called _axioms_ of a _vector space_ (which we'll call the vector space +$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(, 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 $(A_(j,k))^T = A_(k,j)$. diff --git a/notes/2024/math-4a/week-2/refs.bib b/notes/2024/math-4a/week-2/refs.bib new file mode 100644 index 0000000..ef367c4 --- /dev/null +++ b/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}}, +}

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