Linear Algebra Symbols

Category Theory

Symbol	Reference	Other Notation	Remarks
$\to$	arrow/morphism
$\rightarrowtail$	Monic Arrow / Monomorphism
$\twoheadrightarrow$	Epic Arrow / Epimorphism
	Bimorphism		”They exist, but are not studied extensively due to them beign kinda useless”
$\leftrightarrow$	Isomoprhism	$\stackrel{\sim }{\to }$
$f^{-1}$	Inverse of an arrow
$\hookrightarrow$	Inclusion arrow

Set Theory

Symbol	Reference	Other Notation	Latex	Remarks
$\in$	Set membership		\in
$\notin$	Non-set membership		\not\in
$\subseteq$	Subset	$\subset$	\subseteq
$⊊$	Proper Subset	$\subset$	\subsetneq
$\not\subset$	Not a subset		\not\subset
$\forall$	For all elements		\forall
$\exists$, $!\exists$	Exists, unique exist		\exists, !\exists
$S$	Arbitrary Set	$X,Y,U$
$Fun(X,Y)$	The set of all functions from the set $X\to Y$
$X\times Y$	The cartesian product, where $X\times Y=\{(x,y):x\in X,y\in Y\}$		X \times Y
$X\cup Y$	Set union		X \cup Y
$X\cap Y$	Set Intersection		X \cap Y
$\mid S\mid$	Cardinality of a set	$\#S$	\mid S \mid
$\mathbb{N}$	The set of integers starting at 1		\mathbb{N}
$[n]$	The set of all positive integers from 1 to n, $[n]\subset \mathbb{N}$			refer to 1
${\mathbb{N}}_0$	The set of integers starting at 0		\mathbb{N}_0
$\mathbb{Z}$	The set of integers		\mathbb{Z}
$\mathbb{Q}$	The set of rational numbers : $\mathbb{Q}=\{\frac{a}{b}\mid a,b\in \mathbb{Z},b\ne 0\}$		\mathbb{Q}
$\mathbb{R}$	The set of real numbers : $\mathbb{R}$		\mathbb{R}
$\mathbb{C}$	The set of complex numbers : $\mathbb{C}=\{a+bi\mid a,b,\in \mathbb{R}\}$	$\mathbb{R}[i]$	\mathbb{C}
${\aleph }_0$	The first level of countable sets. Ex) : $\mid \mathbb{N}\mid ={\aleph }_0$		\aleph_0
${\aleph }_1$	THe first level of uncountable sets. Ex) : $\mid \mathbb{R}\mid ={\aleph }_1$		\aleph_1	refer to 2

Linear Algebra

Symbol	Reference	Other Notation	Latex	Remarks
$\phi$	Homomorphism between two algebraic structures	$\varphi$	\varphi	The structures will be dependent on the problem
$G$	Group
$R$	Ring
$R[x]$	Ring of Polynomials / Polynomial Ring : $\{{\sum }_{i=0}^n{r}_i{x}^i\mid \forall n\in {\mathbb{N}}_0\}$			In Linear algebra, we will mainly focus on fields, hence $F[x]$
${\mathbb{F}}_n$	Finite Field of n elements		\mathbb{F}
$F$	Field	$K,\mathbb{F}$		The $\mathbb{F}$ will be reserved for free groups, if this book ever covers it
$a,a_i,b,b_i$	An arbitrary element in a field F, $a,b\in F$. Often known as scalars in Lin. Alg.			I will try to use this to denote scalars
$F[\alpha ]$	Field Extension : $F[a]=\{a+b\alpha \mid a,b\in F\}$
F.V.S./ FVS	Vector Space over a field F, F-Vector Space			This is shorthand that may show up throughout the notes
$V,U,W$	Vector Space			Usually, $U,W$ can either mean subspaces of $V$ or entirely different V.S. depending on the problem
$B,C,D$	Basis of Vector Space	$\mathcal{B},\mathcal{C},\mathcal{D},S,\mathcal{S}$
$v_i$	ith basis element in vector space V			Refer to 3 to see indexing methodology
$e_i$	Standard basis element, $e_i=(0,...,1,...,0)$, where the 1 is located on the ith entry
$SB$	Short hand for standard Basis where : $SB=\{e_i\}$
$[\cdot {]}_B$	Coordinate vector with respect to basis $B$			Refer to 4
${\delta }_i$	Delta function
$T,S$	Linear Transformations	$f,F,\mathbb{T}$
$Ho{m}_F(V,W)$	The set of all linear Transformations from t	$L(V,W)$		Reference Below, the F may sometimes be omitted if context is clear
$V^{\vee }$	The Dual Space : $Ho{m}_F(V,F)$	$V^{\ast },V^{\prime }$	V^\vee	Thanks Keith Conrad
$v^{\vee },v_i^{\vee }$	linear functional from the dual space/the ith linear functional in the dual basis	$\varphi ,{\varphi }_i,f_i$	v^\vee
$V^{-\vee }$	The “Anti-Dual”/The set of all left inverses of elements in $V^{\vee }$		V^{-\vee}
$v^{-\vee },v_i^{-\vee }$			v^{-\vee}, V^{-\vee}_i
$V^0$	The set of annihilator Functions	$U^0,W^0$		Some texts may use U, W to make it clearer that its a subspace
$B(V,W)$	Bi-Linear Functionals from $V\times W\to F$			This may also be referred to as Bi-Linear Forms refer to 5
$V\oplus W$	Direct Sum
$V\otimes W$	Direct Product/Tensor Product of FVS
$v\otimes w$	Tensor Product

Homomorphism Mappings

Symbol	Reference
$Hom(X,Y)$	The set of all homomorphisms $\phi$ from $X$ to $Y$. Note, the group structure must be inferred from the problem itself.
$Iso(X,Y)$	The set of all Isomorphisms from $X$ to $Y$
$End(X)$	The set of all endmorphisms : all homomorphisms where $\phi :X\to X$. Note $End(X)=Hom(X,X)$
$Aut(X)$	The set of all automorphisms
Here is a cool mapping on how all of these above are related :

References

1. $[n]$ Adapted from combinatorics, I will denote a subset of positive integers as $[n]$. This is great shorthand as I can express the concept of $1\le i\le n$ as $i\in [n]$ which is shorter to write in latex, is shorter syntactically, and just looks a little cleaner. In come scenarios this notation makes sense as sometimes we want i to be a rational number. Then we can extend this to $i\in [a,b]$. This just seems so natural. One possible problem may occur in confusing this with the change of coordinate vector : $[\cdot {]}_B$ but one must not confuse the two as one has the subscript with the basis, while the other does not!
2. Uncountable Infinity
   1. There are debates as to the correct way to define this. I wont really use this, but it exists. Feel free to read more texts to learn about the history of uncontable infinity
3. Index Notation
   1. In some cases, we may want to deal with more than 2 vector spaces, suppose we have n F.V.S, $V_1,...,V_n$, in the ith FVS, lets say we want to examine the jth basis element. We will denote it as $v_{ij}$. This system will follow similarly for the duals and their functionals
   2. Notation: Over the book, I used $v_1,v_2$ to usually denote the basis vectors is an ordered set, however, i will transition now to a more general term, where $v_{ij}$ will denote a vector from space vector space $V_i$ and the $jth$ basis vector from vector space i. This will expand when we have tensors, and $v_{ijk}$ will eventually become a thing ;-;
4. Dual Space : $V^{\vee }$
   • https://kconrad.math.uconn.edu/blurbs/linmultialg/bilinearform.pdf
   • I really enjoy his use of a dual space as $V^{\vee }$ as opposed to the current notations : $V^{\ast }$ or $V^{\prime }$. While the former has a problem when adressing adjoints and the latter has a problem with derivatives, integral, all other notations in algebra, or sometimes $V$ and $V^{\prime }$ may denote two different vector spaces. I will use $V^{\vee }$ for dual space.
5. $B(V,F)$ : Bilinear forms → Bilinear Functionals
   • Years, modular forms, smooth manifolds, 0-folds, etc. I like Bilinear functionals a little bit more due to its more detailed nature…
6. I want to expand on the notions of the Theory of Vector Spaces through the powerful discovery of category theory. Many contemporary text often rely on the use of set theory, however one consequence of this is the lack of a notion to include the basis of two vector spaces(and in some cases, being the same vector space)
   • For example, suppose we examined the vector space ${\mathbb{R}}^2$. We can equip it with the following basis : $SB=\{e_1,e_2\}$ or $B=\left[\begin{array}{c}1\\ 2\end{array}\right],\left[\begin{array}{c}2\\ 1\end{array}\right]$. I will introduce the following notation: ${\mathbb{R}}_{SB}^2\to {\mathbb{R}}_B^2$ To mean a change of basis, which is a lot clearer than compared to the set theoretic notation ${\mathbb{R}}_{SB}^2\subset {\mathbb{R}}_B^2$
7. Notations jumble: Notation Failure
   • Bad notation : $a^1$ could mean exponent or array, so we try to avoid it

Bibliography

1. Paul Halmos : Naive Set Theory
2. James Munkres : Topology
3. Sheldon Axler : Linear Algebra Done Right
4. Keith Conrad Various Notes