Linear Algebra 1: Vector Spaces

Vector Spaces

A vector space $V$ is a set of objects which can be added and multiplied by numbers, in such a way that the sum of two elements of $V$ is again an element of $V$, the product of an element of $V$ by a number is an element of $V$, and the following properties are satisfied:

VS 1. Given $u,v,w\in V$, we have
$$(u+v)+w=u+(v+w).$$
 

VS 2. There is an element $O\in V$ such that
$$O+u=u+O=u$$
for all $u\in V$.
 

VS 3. Given $u\in V$, the element $(-1)u$ is such that
$$u+(-1)u=(-1)u+u=O.$$
$(-1)u$ is simply written as $-u$.
 

VS 4. For all $u,v\in V$, we have
$$u+v=v+u.$$

VS 5. If $c$ is a number, then $c(u+v)=cu+cv$.
 

VS 6. If $a,b$ are two numbers, then $(a+b)v=av+bv$.
 

VS 7. If $a,b$ are two numbers, then $(ab)v=a(bv)$.
 

VS 8. For any $u\in V$, we have $1u=u$.

The axioms VS 1-VS 4 say that $(V,+)$ is an abelian group. The elements of vector space $V$ are called vectors. One can easily verify that vectors in $\mathbb{R}^n$ satisfy the axioms VS 1-VS 8 and hence $\mathbb{R}^n$ is a vector space.

Example. Let $M(m,n)$ denote the set of all $m\times n$ matrices. Then $M(m,n)$ is a vector space. Using the identification
$$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots &\vdots& \ddots&\vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} \longleftrightarrow(a_{11},\cdots,a_{1n};a_{21},\cdots,a_{2n};\cdots;a_{m1},\cdots,a_{mn}),$$
we see that $M(m,n)$ may be identified with $\mathbb{R}^{mn}$ as a vector space.

Example. Let $\mathcal{F}$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$. For any $f,g\in\mathcal{F}$, define $f+g$ by
$$(f+g)(x)=f(x)+g(x)$$
for all $x\in\mathbb{R}$. For any $f\in\mathbb{R}$ and for any number $c$, define $cf$ by
$$(cf)(x)=cf(x)$$
for all $x\in\mathbb{R}$. Then $\mathcal{F}$ is a vector space called a function space.

Example. Let $V=\{ae^t+be^{2t}: a,b\in\mathbb{R}\}$. Then $V$ is a vector space. Note that $V$ is the set of all solutions of the second order linear differential equation $\frac{d^2x}{dt^2}-3\frac{dx}{dt}+2x=0$.

Subspaces

A subset $U$ of a vector space $V$ is said to be a \emph{subspace} if $U$ itself is also a vector space. For $U$ to be a vector space, it suffices to satisfy that

1. For any $v,w\in U$, $v+w\in U$.
2. If $v\in U$ and $c$ is a number, $cv\in U$.
3. The identity element $O$ of $V$ is also an element of $U$.
   

Proposition. A nonempty subset $U$ of a vector space $V$ is a subspace if and only if $av+bw\in U$ for any $v,w\in U$ and numbers $a,b$.

Proof. Exercise

Example. Let $U$ be the set of vectors in $\mathbb{R}^n$ whose last coordinate is $0$. Then $U$ is a subspace of $\mathbb{R}^n$. $U$ may be identified with $\mathbb{R}^{n-1}$.

Example. Let $A$ be a vector in $\mathbb{R}^n$. Let $U$ be the set of all vectors $B$ in $\mathbb{R}^n$ such that $B\cdot A=0$ i.e. $B$ is perpendicular to $A$. Then $U$ is a subspace of $V$.

Example. Let $U$ and $W$ be subspaces of a vector space $V$. Then $U\cap W$ is also a subspace of $V$.

Example. Let $U$ and $W$ be subspaces of a vector space $V$. Define the sum of $U$ and $W$
$$U+W=\{u+w: u\in U\ \rm{and}\ w\in W\}.$$
Then $U+W$ is a subspace.

Group Theory 4: Examples of Groups

 In here, we defined a group. In here, we have also seen an example of a group, which is a symmetric group. In this lecture note, we study some well-known examples of groups of finite and infinite orders. Recall that the order of a group is the number of elements in the group.

Examples of Groups

1. $(\mathbb{Z},+)$, $(\mathbb{Q},+)$, $(\mathbb{R},+)$, and $(\mathbb{C},+)$ are abelian groups of infinite order.
2. $(\mathbb{Q}\setminus\{0\},\cdot)$ is an abelian group of infinite order.
3. $(\mathbb{R}^+,\cdot)$ is an abelian group of infinite order. Here $\mathbb{R}^+$ denotes the set of all positive real numbers.
4. Let $E_n=\{e^{\frac{2k\pi i}{n}}: k=0,1,2,\cdots,n-1\}$. $e^{\frac{2k\pi i}{n}}$, $k=0,1,2,\cdots,n-1$ are the $n$-th roots of unity i.e. the zeros of $z^n=1$. $E_n$ forms an abelian group of order $n$. It is generated by a single element $e^{\frac{2\pi i}{n}}$. Such a group is called a cyclic group. See the following picture.

 

The cyclic group $E_6$

Definition. A group $G$ is said to be a finite group if it has a finite number of elements. The number of elements in $G$ is called the order of $G$ as mentioned previously and it is denoted by $|G|$ or $\mathrm{ord}(G)$. We use the notation $|G|$ for the order of the group $G$.

Example. Let $\mathbb{Z}_n=\{0,1,2,\cdots,n-1\}$. $\mathbb{Z}_n$ is the set of all possible remainders when an integer is divided by 2. The addition $+$ on $\mathbb{Z}_n$ is defined naturally as follows.
$$\begin{aligned} \mathbb{Z}_2\\ \begin{array}{|c||c|c|} \hline                + & 0 & 1\\                \hline                \hline                0 & 0 & 1\\                \hline                1 & 1 & 0\\                \hline               \end{array} \end{aligned}$$
$$\begin{aligned} \mathbb{Z}_3\\ \begin{array}{|c||c|c|c|} \hline                + & 0 & 1 & 2\\                \hline                \hline                0 & 0 & 1 & 2\\                \hline                1 & 1 & 2 & 0\\                \hline                2 & 0 & 1 & 2\\                \hline               \end{array} \end{aligned}$$
$(\mathbb{Z}_n,+)$ is an abelian group of order $n$. We will see later that the group $E_n$ and $\mathbb{Z}_n$ are indeed the same group.

Example. The set $M_{m\times n}(\mathbb{R})$ of all $m\times n$ matrices of real entries under matrix addition is an abelian group of infinite order.

Example. The set $\mathrm{GL}(n,\mathbb{R})$ of all $n\times n$ non-singular matrices of real entries under matrix multiplication is a non-abelian group of infinite order. $\mathrm{GL}(n,\mathbb{R})$ is called the general linear group of degree $n$. It can be viewed as the set of all linear isomorphisms $T: \mathbb{R}^n\longrightarrow\mathbb{R}^n$ (i.e. linear, one-to-one and onto, but remember from linear algebra that a linear map $T: \mathbb{R}^n\longrightarrow\mathbb{R}^n$ is one-to-one, it is also onto)

Example. [Klein four-group, Vierergruppe in German]

Klein four-group

Let $V=\{e,a,b,c\}$ where $a$ is counterclockwise rotation of the rectangle in the picture about the $x$-axis by $180^\circ$, $b$ is counterclockwise rotation of the rectangle about the $y$-axis by $180^\circ$ and $c$ is counterclockwise rotation of the rectangle about the $z$-axis (the axis coming out of the origin toward you) by $180^\circ$. Then $a$, $b$ and $c$ satisfy the following relationship
$$\begin{align*} a^2=b^2=c^2=e,\ ab&=ba=c,\ bc=cb=a,\\ ca&=ac=b. \end{align*}$$
Here, $\cdot$ is the function composition i.e. successive application of rotations. By labling the vertices of the rectangle as 1, 2, 3, 4 as seen in the picture, we can define each rotation as a permutation of $\{1,2,3,4\}$.
$$\begin{align*} e&=\begin{pmatrix}     1 & 2 & 3 & 4\\     1 & 2 & 3 & 4    \end{pmatrix},\ a=\begin{pmatrix}                       1 & 2 & 3 & 4\\                       2 & 1 & 4 &3                      \end{pmatrix},\\                      b&=\begin{pmatrix}                          1 & 2 & 3 & 4\\                          4 & 3 & 2 & 1                         \end{pmatrix},\ c=\begin{pmatrix}                         1 & 2 & 3 & 4\\                         3 & 4 & 1 &2                         \end{pmatrix}. \end{align*}$$
For instance, we calculate $ab$:
$$\begin{align*}  ab&=\begin{pmatrix}                       1 & 2 & 3 & 4\\                       2 & 1 & 4 &3                      \end{pmatrix}\begin{pmatrix}                          1 & 2 & 3 & 4\\                          4 & 3 & 2 & 1                         \end{pmatrix}\\                         &=\begin{pmatrix}                       1 & 2 & 3 & 4\\                       2 & 1 & 4 &3                      \end{pmatrix}\begin{pmatrix}                      4 & 3 & 2 & 1\\                      3 & 4 & 1 & 2                      \end{pmatrix}\\                      &=\begin{pmatrix}                         1 & 2 & 3 & 4\\                         3 & 4 & 1 &2                         \end{pmatrix}\\                         &=c. \end{align*}$$

Example. [The $n$-th dihedral group $D_n$, $n\geq 3$]
Consider an equilateral triangle shown in the following picture.

Dihedral group $D_3$

$\rho$ is counterclockwise rotation about the axis coming out of the origin by $\frac{360^\circ}{3}=120^\circ$ and $\mu_i$, $i=1,2,3$ is counterclockwise rotation about the axis of rotation through each vertex $i$ by $180^\circ$. As permutations of the vertices $\{1,2,3\}$, $\rho$ and $\mu_i$ , $i=1,2,3$ are given by
$$\begin{align*}  \rho&=\begin{pmatrix}         1 & 2 & 3\\         2 & 3 & 1        \end{pmatrix},\ \mu_1=\begin{pmatrix}        1 & 2 & 3\\        1 & 3 & 2 \end{pmatrix}\\ \mu_2&=\begin{pmatrix}         1 & 2 & 3\\         3 & 2 & 1        \end{pmatrix},\ \mu_3=\begin{pmatrix}        1 & 2 & 3\\        2 & 1 &3 \end{pmatrix}. \end{align*}$$
$D_3=\{e,\rho,\rho^2,\mu_1,\mu_2,\mu_3\}$ is a group called the 3rd dihedral group. $$\mu_1\mu_2=\rho^2\ne\rho=\mu_2\mu_1.$$
So, we see that $D_3$ is not an abelian group. $D_3$ is the group of symmetries of an equilateral triangle.

Now this time let us consider a square as shown in the following picture.

Dihedral group $D_4$

$\rho$ is counterclockwise rotation about the $z$-axis coming out of the origin by $\frac{360^\circ}{4}=90^\circ$. $\mu_i$, $i=1,2$ are counterclockwise rotations about $y$-axis and $x$-axis, respectively by $180^\circ$. $\delta_i$, $i=1,2$ are counterclockwise rotations about the axis through the vertices 2 and 4 and the vertices through 1 and 3, respectively by $180^\circ$. As permutations of the vertices $\{1,2,3,4\}$, $\rho$, $\mu_i$ and $\delta_i$, $i=1,2$ are given by
$$\begin{align*}  \rho&=\begin{pmatrix}         1 & 2 & 3 & 4\\         2 & 3 & 4 & 1        \end{pmatrix},\\        \mu_1&=\begin{pmatrix}                1 & 2 & 3 & 4\\                2 & 1 & 4 & 3               \end{pmatrix},\ \mu_2=\begin{pmatrix}               1 & 2 & 3 & 4\\               4 & 3 & 2 & 1 \end{pmatrix},\\ \delta_1&=\begin{pmatrix}            1 & 2 & 3 & 4\\            3 & 2 & 1 & 4           \end{pmatrix},\ \delta_2=\begin{pmatrix}           1 & 2 & 3 & 4\\           1 & 4 & 3 & 2           \end{pmatrix}. \end{align*}$$
$D_4=\{e,\rho,\rho^2,\rho^3,\mu_1,\mu_2,\delta_1,\delta_2\}$ is the group of symmetries of a square, called the 4th dihedral group. It is also called the octic group. Note that $|D_n|=2n$, $n\geq 3$.

Group Theory 3: Basic Number Theory

 In this lecture note, we study some basic number theory as it is needed to study group theory.

Let $\mathbb{Z}$ denote the set of integers. $\mathbb{Z}$ satisfies well-ordering principle, namely any non-empty set of nonnegative integers has a smallest member.

One of the most fundamental theorems regarding numbers is Euclid's Algorithm. Although we will not discuss its proof, it can be proved using well-ordering principle.
Theorem. [Euclid's Algorithm] If $m$ and $n$ are integers with $n>0$, then $\exists$ integers $q$ and $r$ with $0\leq r<n$ such that $m=qn+r$.

Euclid's algorithm hints us how we can define the notion that one integer divides another.

Definition. Given $m\ne 0, n\in\mathbb{Z}$, we say $m$ divides $n$ and write $m|n$ if $n=cm$ for some $c\in\mathbb{Z}$.


Example. $2|14$, $(-7)|14$, $4|(-16)$.

If $m|n$, we call $m$ a divisor or a factor of $n$, and $n$ a multiple of $m$. To indicate $m$ is not a divisor of $n$, we write $m\not|n$. For example, $3\not|5$.

Lemma. The following properties hold.

1. $1|n$ $\forall n$.
2. If $m\ne 0$ then $m|0$.
3. If $m|n$ and $n|q$, then $m|q$.
4. If $m|n$ and $m|q$ then $m|(\mu n+\nu q)$ $\forall \mu,\nu$.
5. If $m|1$ then $m=\pm 1$.
6. If $m|n$ and $n|m$ then $m=\pm n$.
   

Definition. Given $a,b$ (not both 0), their greatest common divisor (in short gcd) $c$ is defined by the following properties:

1. $c>0$
2. $c|a$ and $c|b$
3. If $d|a$ and $d|b$ then $d|c$.
   

If $c$ is the gcd of $a$ and $b$, we write $c=(a,b)$.

$(24,9)=3$. Note that the gcd 3 can be written in terms of 24 and 9 as $3\cdot 9+1\cdot (-24)$ or $(-5)9+2\cdot 24$. In general, we have the following theorem holds.

Theorem. If $a,b$ are not both 0, their gcd exists uniquely. Moreover, $\exists m,n\in\mathbb{Z}$ s.t. $c=ma+nb$.

Now let us talk about how to find the gcd of two positive numbers $a$ and $b$. W.L.O.G. (Without Loss Of Generality), we may assume that $b<a$. Then by Euclid's algorithm we have
$$a=bq+r,\ \mbox{where}\ 0\leq r<b.$$
Let $c=(a,b)$. Then $c|r$, so $c$ is a common divisor of $b$ and $r$. If $d$ is a common divisor of $b$ and $r$, it is also a common divisor of $a$ and $b$. This implies that $d\leq c$ and so $c=(b,r)$. Finding $(b,r)$ is of course easier because one of the numbers is smaller than before.
Example. [Finding GCD]
$$\begin{aligned}  (100,28)&=(28,16)\ &(100&=28\cdot 3+16)\\  &=(16,12)\ &(28&=16\cdot 1+12)\\  &=(12,4)\ &(16&=12\cdot 1+14)\\  &=4. \end{aligned}$$
By working backward, we can also find integers $m$ and $n$ such that
$$4=m\cdot 100+n\cdot 28.$$
$$\begin{align*}  4&=16+12(-1)\\  &=16+(-1)[28+(-1)16]\\  &=(-1)28+2\cdot 16\\  &=(-1)28+2[100+(-3)28]\\  &=2\cdot 100+(-7)28. \end{align*}$$
Therefore, $m=2$ and $n=-7$.

Definition. We say that $a$ and $b$ are relatively prime if $(a,b)=1$.

Theorem. The integers $a$ and $b$ are relatively prime if and only if $1=ma+nb$ for some $m$ and $n$.

Theorem. If $(a,b)=1$ and $a|bc$ then $a|c$.

Theorem. If $b$ and $c$ are both relatively prime to $a$, then $bc$ is also relatively prime to $a$.

Definition. A prime number, or shortly prime, is an integer $p>1$ such that $\forall a\in\mathbb{Z}$, either $p|a$ or $(p,a)=1$.

Suppose that $p$ is a prime as defined above and $p=ab$, where $1\leq a<p$. Then $p\not|a$ since $a<p$, so $(p,a)=1$. This implies that $p|b$. On the other hand, $b|p(=ab)$ and hence $p=b$ and $a=1$. So, the above definition coincides with the definition of a prime we are familiar with.

Theorem. If $p$ is a prime and $p|a_1a_2\cdots a_n$, then $p|a_i$ for some $i$ with $1\leq i\leq n$.

Proof. If $p|a_1$, we are done. If not, $(p,a_1)=1$ and so $p|a_2a_3\cdots a_n$. Continuing this, we see that $p|a_i$ for some $i$.

Regarding primes, we have the following theorems.

Theorem. If $n>1$, then either $n$ is a prime or the product of primes.

Theorem. [Unique Factorization Theorem] Given $n>1$, there is a unique way to write $n$ in the form $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, where $p_1<p_2<\cdots<p_k$ are primes and the exponents $a_1,\cdots,a_k$ are all positive.

Theorem. [Euclid] There is an infinite number of primes.

Group Theory 2: Functions

 Let me first mention some logical symbols I will use often. They are $\forall$ which means ``for all'', ``for any'', ``for each'', or ``for every'' depending on the context, $\exists$ which means ``there exists'', and $\ni$ which means ``such that'' (don't be confused with $\in$ which means ``be an element of''). We also use s.t. for ``such that.'' There are also $\Longrightarrow$ which means ``implies'' and $\Longleftrightarrow$ which means ``if and only if.'' I guess these pretty much cover what we use most of time.

Now let us review functions in a more formal way. Let $X$ and $Y$ be two non-empty sets. The the Cartesian product $X\times Y$ of $X$ and $Y$ is defined as the set
$$X\times Y=\{(x,y): x\in X,\ y\in Y\}.$$
A subset $f$ of the Cartesian product $X\times Y$ (we write $f\subset X\times Y$) is called a graph from $X$ to $Y$. A graph $f\subset X\times Y$ is called a function from $X$ to $Y$ (we write $f: X\longrightarrow Y$) if whenever $(x,y_1),(x,y_2)\in f$, $y_1=y_2$. If $f: X\longrightarrow Y$ and $(x,y)\in f$, we also write $y=f(x)$. A function $f: X\longrightarrow Y$ is said to be one-to-one or injective if whenever $(x_1,y),(x_2,y)\in f$, $x_1=x_2$. This is equivalent to saying $f(x_1)=f(x_2)$ implies $x_1=x_2$. A function $f: X\longrightarrow Y$ is said to be onto or surjective if $\forall y\in Y$ $\exists x\in X$ s.t. $(x,y)\in f$. A function $f: X\longrightarrow Y$ is said to be one-to-one and onto (or bijective) if it is both one-to-one and onto (or both injective and surjective).

Let $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ be two functions. Then the composition or the composite function $g\circ f: X\longrightarrow Z$ is defined by $g\circ f(x)=g(f(x))$ $\forall x\in X$. The function composition $\circ$ may be considered as an operation and it is associative.

Lemma. If $h: X\longleftrightarrow Y$, $g:Y\longleftrightarrow Z$ and $f:Z\longleftrightarrow W$, then $f\circ(g\circ h)=(f\circ g)\circ h$.

Note that $\circ$ is not commutative i.e. it is not necessarily true that $f\circ g=g\circ f$ even when both $f\circ g$ and $g\circ f$ are defined.

The following lemmas will be useful when we study group theory later.

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are one-to-one, so is $g\circ f: X\longrightarrow Z$.

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are onto, so is $g\circ f: X\longrightarrow Z$.

As an immediate consequence of combining these two lemmas, we obtain

Lemma. If both $f: X\longrightarrow Y$ and $g: Y\longrightarrow Z$ are bijective, so is $g\circ f: X\longrightarrow Z$.

If $f\subset X\times Y$, then the inverse graph $f^{-1}\subset Y\times X$ is defined by
$$f^{-1}=\{(y,x)\in Y\times X: (x,y)\in f\}.$$
If $f: X\longrightarrow Y$ is one-to-one and onto (bijective) then its inverse graph $f^{-1}$ is a function $f^{-1}: Y\longrightarrow X$. The inverse $f^{-1}$ is also one-to-one and onto.

Lemma. If $f: X\longrightarrow Y$ is a bijection, then $f\circ f^{-1}=\imath_Y$ and $f^{-1}\circ f=\imath_X$, where $\imath_X$ and $\imath_Y$ are the identity mappings of $X$ and $Y$, respectively.

Let $A(X)$ be the set of all one-to-one functions of $X$ onto $X$ itself. Then $(A(X),\circ)$ is a group. If $X$ is a finite set of $n$-elements (we may conveniently say $X=\{1,2,\cdots,n\})$, then $(A(X),\circ)$ is a finite group of order $n!$, called the symmetric group of degree $n$. The symmetric group of degree $n$ is denoted by $S_n$ and the elements of $S_n$ are called permutations.

Group Theory 1: An Overview

Before we begin to discuss the main subject, I would like to give an overview of what we study in group theory or more generally in algebra.

Algebra (as a subject) is the study of algebraic structures. So, what is an algebraic structure? An algebraic structure or an algebra in short $\underline{A}$ is a non-empty set $A$ with a binary operation $f$. (In general, a set of operations but here I consider only one binary operation for simplicity.) $\underline{A}$ is usually written as the ordered pair
$$\underline{A}=(A,f).$$
A binary operation $f$ on a set $A$ is a function $f: A\times A\longrightarrow A$. An example of a binary operation is addition $+$ on the set of integers $\mathbb{Z}$. $+$ is a function $+:\mathbb{Z}\times\mathbb{Z}\longrightarrow\mathbb{Z}$ defined by $+(1,1)=2$, $+(1,2)=3$, and so on. We usually write $+(1,1)=2$ as $1+1=2$. In general, one may consider an $n$-ary operation $f:\prod_{i=1}^n A\longrightarrow A$, where $\prod_{i=1}^n A$ denotes the $n$-copies of $A$, $A\times A\times\cdots\times A$.

There are many different kinds of algebras. Let me mention some of algebras with a binary operation here. For starter, $(A,\cdot)$, a non-empty set $A$ with a binary operation $\cdot$ is called a groupoid. A groupoid $(A,\cdot)$ with associative law
$$(ab)c=a(bc)$$
for any $a,b,c\in A$ is callaed a semigroup. If the semigroup has an identity element $e\in A$ i.e.
$$ae=ea=a$$
for any $a\in A$, it is called a monoid. If for every element $a$ of the monoid $A$, there exists an inverse element $a^{-1}\in A$ such that $aa^{-1}=a^{-1}a=e$, the monoid is called a group. A group $(A,\cdot)$ with commutative law i.e.
$$ab=ba,$$
for any $a,b\in A$ is called an abelian group named after a Norwegian mathematician Niels Abel. Note the inverse ${}^{-1}$ can be regarded as an operation on $A$, a unary operation ${}^{-1}: A\longrightarrow A$ defined by ${}^{-1}(a)=a^{-1}$ for each $a\in A$. The identity element $e$ can be also regarded as an operation, a nullary operation $e:\{\varnothing\}\longrightarrow A$. Thus, formally a group can be written as $(A,\cdot,{}^{-1},e)$, a quadrupple of a nonempty set, a binary operation, a unary operation, and a nullary operation.

Now we know what a group is and apparently, group theory is the study of groups. But what exactly are we studying there? What I am about to say is not really limited to group theory but commonly applies to studying other algebraic structures as well. There are briefly two main objectives with studying groups. One is the classification of groups. This becomes particularly interesting with groups of finite order. Here the order of a group means the number of elements of a group. We would like to answer the question ``how many different groups of order $n$ are there for each $n$ and what are they?'' The classification gets harder as $n$ gets larger. There are groups with the same order that appear to be different. But don't be deceived by the appearance. They may actually be the same group. What do we mean by same here? We say two groups of the same order same if there is a one-to-one and onto map (a bijection) that preserves operations. Such a map is called an isomorphism. It turns out that if a map $\psi: G\longrightarrow G'$ from a group $G$ to another group $G'$ preserves binary operation, it automatically preserves unary and nullary operations. Here we mean preserving binary operation by
$$\psi(ab)=\psi(a)\psi(b)$$
for any $a,b\in G$. A map $\psi: G\longrightarrow G'$ which preserves binary operation is called a homomorphism. If a homomorphism $\psi: G\longrightarrow G'$ is one-to-one and onto, it is an isomorphism. An isomorphism $\psi: G\longrightarrow G$ from a group $G$ onto itself is called an automorphism. In group theory, if there is an isomorphism from a group to another group, we do not distinguish them no matter how different they appear to look. The other objective in studying group theory is to discover new groups from old groups. Some of the new groups may be smaller in size than the old ones. Here we mean smaller in size by having a smaller number of elements i.e. having a lesser order. Some examples are subgroups and quotient groups. Some of the new groups are larger in size than the old ones. An example is direct products. Subgroups, quotient groups (also called factor groups), direct products are the things we will study as means to get new groups from old groups.

Group theory has great significance in geometry. In geometry, symmetry plays an important role. There are different types of symmetries: reflections, rotations, and translations. An interesting connection between geometry and group theory is that these symmetries form a group. The most general such group of finite order is called a symmetric group. In mathematics, the embedding theorem is conceptually and philosophically important, although it may be practically less important. When we study mathematics, we often feel that the structures we study are highly abstract and we feel like they only exist in our consciousness but not in the physical world. The embedding theorem tells that those abstract structures we study are indeed substructures of a larger structure that we are familiar with in the physical world. The embedding theorem implicates that we are not making up those abstract mathematical structures but we are merely discovering them which already exist in the universe. This kind of view point is called Mathematical Platonism. It turns out that there is an embedding theorem in finite group theory, namely every group of finite order is a subgroup of a symmetric group. The embedding theorem is called Cayley theorem. This means that the study of finite groups boils down to studying symmetric groups.

Front for Mathphys Archive

 "I write not because I know something but to learn something."

"The most important book to me is my own notebook because it is written in the way I understood."

This pinned post contains the list of my lecture notes on mathematics, physics, and related areas (theoretical computer science, mathematical biology, and mathematical finance). They are categorized by subjects in each area for readers' convenience. This post is subject to updates as I continue to add new lecture notes.

Mathematics

Advanced Calculus

Lecture Notes

Problem Sets

References and Further Reading

Ivan S. Sokolnikoff, Advanced Calculus, McGraw-Hill Book Compny, Inc., 1939

Abstract Algebra I: Groups

Lecture Notes

Group Theory 1: An Overview

Group Theory 2: Functions

Group Theory 3: Basic Number Theory

Group Theory 4: Examples of Groups 

Problem Sets

Group Theory Problem Set 1: Basic Number Theory

Group Theory Problem Set 2: Examples of Groups

References and Further Reading

John B. Fraleigh, A First Course in Abstract Algebra, 7th Edition, Pearson, 2002
I. N. Herstein, Abstract Algebra, 3rd Edition, Wiley, 1996
I. N. Herstein, Topics in Algebra, 2nd Edition, John Wiley & Sons, 1975
Ramji Lal, Algebra 1, Groups, Rings, Fields and Arithmetic, Springer, 2017

Abstract Algebra II: Rings, Fields, Galois Theory, Modules, Representation Theory

Lecture Notes 

Problem Sets

References and Further Reading

John B. Fraleigh, A First Course in Abstract Algebra, 7th Edition, Pearson, 2002
I. N. Herstein, Abstract Algebra, 3rd Edition, Wiley, 1996
I. N. Herstein, Topics in Algebra, 2nd Edition, John Wiley & Sons, 1975
Ramji Lal, Algebra 1, Groups, Rings, Fields and Arithmetic, Springer, 2017
Ramji Lal, Algebra 2, Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier, Springer, 2017

Functional Analysis/Fourier Analysis

Functional Analysis

Lecture Notes 

Problem Sets

Fourier Analysis

Lecture Notes 

Problem Sets

References and Further Reading

Gerald B. Folland, Fourier Analysis and Its Applications, Pure and Applied Undergraduate Texts, American Mathematical Society, 2009
Sigurdur Helgason, Topics in Harmonic Analysis on Homogeneous Space, Birkhäuser
Einer Hille and Ralph S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, 1957
Erwin Kreyszig, Introductory Functional Analysis with Applications, 1st Edition, Wiley, 1989
Michael Reed and Barry Simon, Methods of Mathematical Physics I: Functional Analysis, Revised and Enlarged Edition, Academic Press, 1980
Michael Reed and Barry Simon, Methods of Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1972
Michael Reed and Barry Simon, Methods of Mathematical Physics III: Scattering Theory, Academic Press, 1972
Michael Reed and Barry Simon, Methods of Mathematical Physics IV: Analysis of Operators, Academic Press, 1978
H. L. Royden, Real Analysis, Second Edition, The Macmillan Company

Walter Rudin, Functional Analysis, McGraw-Hil Book Company, 1973
Peter Szekeres, A Course in Modern Mathematical Physics, Groups, Hilbert Space and Differential Geometry, Cambridge University Press, 2004

Functions of a Complex Variable 

Lecture Notes

Problem Sets

References and Further Reading

Lars Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill, 1979

James Brown and Ruel Churchill, Complex Variables and Applications, 8th Edition, McGraw-Hill, 2008

John B. Conway, Functions of One Complex Variable I, 2nd Edition, Graduate Texts in Mathematics, Springer, 1978

Linear Algebra

Lecture Notes

Linear Algebra 1: Vector Spaces

Problem Sets

References and Further Reading

Sheldon Axler, Linear Algebra Done Right, Third Edition, Springer, 2015
Ramji Lal, Algebra 2, Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier, Springer, 2017
Serge Lang, Introduction to Linear Algebra, Second Edition, Springer, 1986
Serge Lang, Linear Algebra, Third Edition, Springer, 2004

Number Theory and Cryptography

Software Tools for Computation

Study of number theory is often accompanied by heavy computations for which computers can be effectively used. From time to time, I will be using Maxima, an open source computer algebra system (abbreviated as CAS, a software package for symbolic computation) and Python. Sage (CoCalc) is also a good and powerful open source CAS but I will not be using it here. Main reason is that I find it restrictive as it requires a web-based interface (a browser) and an internet connection. Unless you run your own CoCalc server (if you are going to use it just for yourself, why would you run such a server?), you need to have an access to a CoCalc server that is run by someone else. I personally prefer a crude and low tech computing environment that does not require a particular interface and a lot of resources. Maxima and Python fit into such preference of mine. Maxima and Python do have interfaces but they also can be run interface-free in a command shell.

MAXIMA

What is Maxima?: MIT has developed a computer algebra system, called Macsyma, from 1968 to 1982 as part of Project MAC. They turned over a copy of the Macsyma source code to the Department of Energy (DOE). That version is known as DOE Macsyma. It had been maintained by William Schelter at the University of Texas from 1982 until his death in 2001. In 1998, Schelter obtained permission from the DOE to release the DOE Macsyma source code under the GNU Public License, and in 2000 he initiated the Maxima project at SourceForge to maintain and continue developing DOE Macsyma, now called Maxima.

 

Download and Install Maxima: 1. For those who are using Windows, an instruction can be found here and also here.
2. For those who are using MacOS, Maxima can be download from here and an installation instruction can be found here.
3. If you are using Ubuntu Linux OS, simply run the following command in command shell:

sudo apt install maxima && sudo apt install wxmaxima

You also need to install Gnuplot by runnig the command:

sudo apt install gnuplot

4. If you are using FreeBSD Unix, as root run:

# pkg install maxima && pkg install wxmaxima

and also install Gnuplot by running

# pkg install gnuplot

5. For all other operating systems, refer to the instruction at Maxima Downloads page here.

Maxima documentation page is an important source on how to use Maxima for your computational needs and purposes. Another important source on using Maxima, especially for doing mathematics is The MaximaList. The html version of Maxima 5.29.0 manual on Number Theory is available online here. 

 PYTHON

You can find download/installtion instruction for your OS at Python.org. Make sure that you install Python 3.

1. For Windows system, select and download Python Windows installer, for example python-3.3.0.msi and run it.
2. For MacOS, select and download .dmg Mac Installer Disk Image and run it.
3. For Ubuntu Linux OS, run in command shell, for example

sudo apt install python3.3 && sudo apt install idle3

IDLE is a covenient interface for Python. I love iPython (Interactive Python Shell). It allows you to run .py files in command shell. To install ipython run

sudo apt install ipython3

4. For FreeBSD Unix, as root run:

# pkg install python

This command will install Python 3 and IDLE 3 as well. To install ipython, as root run:

# pkg install py37-ipython

Lecture Notes

Problem Sets

References and Further Reading

Elementary Number Theory

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Oxford at the Clarendon Press, 1975
Neal Koblitz, A Course in Number Theory and Cryptogtraphy, Graduate Texts in Mathematics 114, Springer-Verlag, 1994
Franz Lemmermeyer, Numbers and Curves, Springer-Verlag, 2001
Manfred Schroeder, Number Theory in Science and Communication with Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, Fifth Edition, Springer-Verlag, 2009
Simon Singh, Fermat's Last Theorem, The Story of a Riddle that Confounded the World's Greatest Minds for 358 Years, Fourth Estate
André Weil, Number Theory for Beginners, Springer-Verlag, 1979

Cryptography

Sara Arias-de-Reyna and Gabor Wiese, Algebraic Curves and Applications to Cryptography
Steven Galbraith, Mathemathematics of Public Key Cryptography
Neal Koblitz, A Course in Number Theory and Cryptogtraphy, Graduate Texts in Mathematics 114, Springer-Verlag, 1994
Gabor Wiese, Théorie des nombres et applications à la cryptographie

Elliptic Curves

J.W.S. Cassels, Lectures on Elliptic Curves, Cambridge University Press, 1991
Dale Husemöller, Elliptic Curves, Graduate Texts in Mathematics 111, Springer, 2002
Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics 97, Springer-Verlag, 1984
J.S. Milne, Elliptic curves, 1996
Bjorn Poonen, Elliptic Curves, 2001
Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1985

Probability

Lecture Notes

Problem Sets

References and Further Reading

Sheldon Ross, A First Course in Probability, Fifth Edition, Prentice Hall, 1997

Physics

Classical Mechanics 

Lecture Notes

Problem Sets

References and Further Reading

A. P. French, Newtonian Mechanics, The M.I.T. Introductory Physics Series, Thomas Nelson and Sons LTD, 1971
Walter Greiner, Classical Mechanics, Point Particles and Relativity, Springer, 2003
Walter Greiner, Classical Mechanics, Systems of Particles and Hamiltonian Dynamics, Springer, 2000
Walter D. Knight and Malvin A Ruderman, Mechanics, Charles Kittel, Berkeley Physics Course Volume 1, Second Edition, McGraw-Hill, 1973
L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics Volume 1, Third Edition, Elsevier, 1976
Jakob Schwichtenberg, No-Nonsense Classical Mechanics, A Student Friendly Introduction, No-Nonsense Books, 2019

Mathematical Physics 

Lecture Notes

Problem Sets

References and Further Reading

George Arfken, Mathematical Methods for Physicists, Third Edition, Academic Press, INC., 1985

Computer Science

Biology

Finance