## Quick overview of algebraic structures This page covers some basic idea of most common algebraic structures, to avoid unfamiliarity when talking about these concepts. ### Group A **group** is a set $G$ equipped with a binary operation $(\cdot)$, such that the following properties hold: 1. Closure: For all $a, b \in G$, $a \cdot b \in G$. 2. Associativity: For all $a, b, c \in G$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. 3. Identity: There exists an element $e \in G$ such that for all $a \in G$, $a \cdot e = a = e \cdot a$. 4. Inverse: For every $a \in G$, there exists an element $a^{-1} \in G$ such that $a \cdot a^{-1} = e = a^{-1} \cdot a$. If the group $G$ also satisfies the property: - Commutativity: For all $a, b \in G$, $a \cdot b = b \cdot a$, then $G$ is called an **abelian group**. > [!Example]- > - The set of non-zero real numbers $\mathbb{R}^*$ under multiplication forms a group but is not a ring or field (since it lacks an additive operation). > - The set of non-zero rational numbers $\mathbb{Q}^*$ under multiplication forms an abelian group but is not a ring or field (since it lacks an additive operation). > - The set of $2 \times 2$ invertible matrices under matrix multiplication forms a group but is not a ring or field (since it lacks closure under addition and additive inverses). ### Ring A **ring** is a set $R$ equipped with two binary operations: addition $(+)$ and multiplication $(\cdot)$, such that the following properties hold: 1. Additive Closure: For all $a, b \in R$, $a + b \in R$. 2. Additive Associativity: For all $a, b, c \in R$, $(a + b) + c = a + (b + c)$. 3. Additive Identity: There exists an element $0 \in R$ such that for all $a \in R$, $a + 0 = a$. 4. Additive Inverse: For every $a \in R$, there exists an element $-a \in R$ such that $a + (-a) = 0$. 5. Additive Commutativity: For all $a, b \in R$, $a + b = b + a$. 6. Multiplicative Closure: For all $a, b \in R$, $a \cdot b \in R$. 7. Multiplicative Associativity: For all $a, b, c \in R$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. 8. Distributivity: For all $a, b, c \in R$, $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ and $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$. If the ring $R$ also satisfies the property: - Multiplicative Identity: There exists an element $1 \in R$ such that for all $a \in R$, $a \cdot 1 = a = 1 \cdot a$, then $R$ is called a **ring with unity**. If multiplication is also commutative: - Multiplicative Commutativity: For all $a, b \in R$, $a \cdot b = b \cdot a$, then $R$ is called a **commutative ring**. > [!Example]- > - The set of integers $\mathbb{Z}$ with the usual addition and multiplication forms a ring (but not a field, since not all elements have a multiplicative inverse). > - The set of $n \times n$ matrices over a field forms a ring under matrix addition and multiplication. ### Field A **field** is a set $F$ equipped with two binary operations: addition $(+)$ and multiplication $(\cdot)$, such that the following properties hold: 1. Additive Closure: For all $a, b \in F$, $a + b \in F$. 2. Additive Associativity: For all $a, b, c \in F$, $(a + b) + c = a + (b + c)$. 3. Additive Identity: There exists an element $0 \in F$ such that for all $a \in F$, $a + 0 = a$. 4. Additive Inverse: For every $a \in F$, there exists an element $-a \in F$ such that $a + (-a) = 0$. 5. Additive Commutativity: For all $a, b \in F$, $a + b = b + a$. 6. Multiplicative Closure: For all $a, b \in F$, $a \cdot b \in F$. 7. Multiplicative Associativity: For all $a, b, c \in F$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. 8. Multiplicative Identity: There exists an element $1 \in F$ such that for all $a \in F$, $a \cdot 1 = a = 1 \cdot a$. 9. Multiplicative Inverse: For every $a \in F \setminus \{0\}$, there exists an element $a^{-1} \in F$ such that $a \cdot a^{-1} = 1$. 10. Multiplicative Commutativity: For all $a, b \in F$, $a \cdot b = b \cdot a$. 11. Distributivity: For all $a, b, c \in F$, $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$ and $(a + b) \cdot c = (a \cdot c) + (b \cdot c)$. > [!Example]- > - The set of rational numbers $\mathbb{Q}$, real numbers $\mathbb{R}$, and complex numbers $\mathbb{C}$ are all fields under the usual addition and multiplication operations. > - The set of integers modulo a prime $p$, denoted $\mathbb{Z}_p$, forms a field with addition and multiplication defined modulo $p$. > [!Note] > - A **group** has only one binary operation, whereas a **ring** has two: addition and multiplication. > - A **ring** extends the concept of a group by adding a second operation (multiplication) that must be distributive over addition. > - A **field** extends the concept of a ring by requiring that all non-zero elements have a multiplicative inverse and that multiplication is commutative. ### Bilinear form A **bilinear form** is a function $B: V \times V \to F$ that is linear in each argument, where $V$ is a vector space over a field $F$. Specifically, for all $u, v, w \in V$ and $a, b \in F$, the following properties hold: 1. Additivity in the First Argument: $B(u + v, w) = B(u, w) + B(v, w)$. 2. Homogeneity in the First Argument: $B(a u, v) = a B(u, v)$. 3. Additivity in the Second Argument: $B(u, v + w) = B(u, v) + B(u, w)$. 4. Homogeneity in the Second Argument: $B(u, b v) = b B(u, v)$. A bilinear form becomes the **inner product** (or **dot product** in Euclidean space) if it additionally satisfies the following properties: 1. Symmetry: $B(u, v) = B(v, u)$ for all $u, v \in V$. 2. Positive Definiteness: $B(v, v) > 0$ for all non-zero $v \in V$. If these conditions are met, the bilinear form is called a **positive definite symmetric bilinear form**, which is equivalent to the dot product in any inner product space (include the Euclidean space). With dot product in an inner product space, we can define the Euclidean norm as lengths, $\| \boldsymbol{x}\| := \sqrt{\boldsymbol{x} \cdot \boldsymbol{x}}$ >[!Notice] >It is worth to mention that **the orthogonality is defined by inner product being zero**. Namely it is >$\mathbf{u}\cdot \mathbf{v}=0$ >gives the orthogonality. >This requires that the inner product is well defined. ### Algebra An **algebra** over a field $F$ is a vector space $A$ over $F$ equipped with a bilinear multiplication operation $(\cdot): A \times A \to A$, such that the following properties hold: 1. **Vector Space Structure**: $A$ is a vector space over $F$. That is, it satisfies closure, associativity, identity, and scalar multiplication under addition. 2. **Multiplicative Closure**: For all $a, b \in A$, $a \cdot b \in A$. 3. **Multiplicative Associativity** (optional): For all $a, b, c \in A$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. 4. **Distributivity over Addition**: - Left: $a \cdot (b + c) = a \cdot b + a \cdot c$ - Right: $(a + b) \cdot c = a \cdot c + b \cdot c$ 5. **Bilinearity over the field $F$**: - $(\lambda a) \cdot b = \lambda (a \cdot b)$ and $a \cdot (\lambda b) = \lambda (a \cdot b)$ for all $a, b \in A$, $\lambda \in F$. If the multiplication is associative, the algebra is called an **associative algebra**. If the multiplication is also commutative: - **Multiplicative Commutativity**: $a \cdot b = b \cdot a$ for all $a, b \in A$, then it is called a **commutative algebra**. If there exists an identity element $1 \in A$ such that $a \cdot 1 = a = 1 \cdot a$ for all $a \in A$, then $A$ is called an **algebra with unity**. > [!Example]- > - The set of $n \times n$ matrices over a field $F$, denoted $M_n(F)$, forms an associative (but generally non-commutative) algebra with unity. > - The polynomial ring $F[x]$ is a commutative algebra over $F$. > - The complex numbers $\mathbb{C}$ form a commutative algebra over the reals $\mathbb{R}$. > [!Note]- > The set of all linear transformations from a vector space $V$ to itself, denoted $\mathrm{End}(V)$, forms an associative algebra over the field $F$. When $V = F^n$, this algebra is isomorphic to the full matrix algebra $M_n(F)$ — the set of all $n \times n$ matrices with entries in $F$. > > $ > \mathrm{End}(F^n) \cong M_n(F) > $ > > This structure is often called the **matrix algebra** or the **endomorphism algebra**, and it plays a central role in representation theory, geometry, and physics. It is associative, has a unity, but is generally non-commutative when $n > 1$.