Added inner product spaces.

config/en/mkdocs.yaml

@@ -88,6 +88,7 @@ nav:
       - 'Determinants': mathematics/linear-algebra/determinants.md
       - 'Vector spaces': mathematics/linear-algebra/vector-spaces.md
       - 'Linear transformations': mathematics/linear-algebra/linear-transformations.md
+      - 'Inner product spaces': mathematics/linear-algebra/inner-product-spaces.md
       - 'Orthogonality': mathematics/linear-algebra/orthogonality.md
       - 'Diagonalization': mathematics/linear-algebra/diagonalization.md
     - 'Calculus': 
@@ -135,7 +136,7 @@ nav:
           - 'Oscillations': physics/mechanics/lagrangian-mechanics/applications/oscillations.md
       - 'Hamiltonian mechanics': 
         - 'Hamiltonian formalism': physics/mechanics/hamiltonian-mechanics/hamiltonian-formalism.md
-        - "Hamilton's equations": physics/mechanics/hamiltonian-mechanics/equations-of-hamilton.md
+        - 'Equations of Hamilton': physics/mechanics/hamiltonian-mechanics/equations-of-hamilton.md
 #      - 'Relativistic mechanics':
 #    - 'Quantum mechanics':
     - 'Electromagnetism': 

docs/en/mathematics/index.md

@@ -11,6 +11,7 @@ Welcome to the mathematics page. Some special mathematical environments that wil
 * *Lemmas* : a minor result whose purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem.
 * *Corollaries* : a result in which the proof relies heavily on a given theorem.
 * *Proofs* : a convincing argument that a certain mathematical statement is necessarily true. A proof generally uses deductive reasoning and logic but also contains some amount of ordinary language.
+* *Conjectures*: a conclusion or a proposition that is proffered on a tentative basis without proof.
 
 The mathematics sections of this wiki are based on various books and lectures. A comprehensive list of references can be found below.
 

docs/en/mathematics/linear-algebra/determinants.md

@@ -4,7 +4,7 @@
 
 With each $n \times n$ matrix $A$ with $n \in \mathbb{N}$ it is possible to associate a scalar, the determinant of $A$ denoted by $\det (A)$ or $|A|$.
 
-> *Definition*: let $A = (a_{ij})$ be an $n \times n$ matrix and let $M_{ij}$ denote the $(n-1) \times (n-1)$ matrix obtained from $A$ by deleting the row and column containing $a_{ij}$ with $n \in \mathbb{N}$ and $(i,j) \in \{1, \dots, n\} \times \{1, \dots, n\}$. The determinant of $M_{ij}$ is called the **minor** of $a_{ij}$. We define the **cofactor** of $A_{ij}$ of $a_{ij}$ by
+> *Definition*: let $A = (a_{ij})$ be an $n \times n$ matrix and let $M_{ij}$ denote the $(n-1) \times (n-1)$ matrix obtained from $A$ by deleting the row and column containing $a_{ij}$ with $n \in \mathbb{N}$ and $(i,j) \in \{1, \dots, n\} \times \{1, \dots, n\}$. The determinant of $M_{ij}$ is called the **minor** of $a_{ij}$. We define the **cofactor** of $a_{ij}$ by
 >
 > $$
 >   A_{ij} = (-1)^{i+j} \det(M_{ij}).

docs/en/mathematics/linear-algebra/inner-product-spaces.md

@@ -0,0 +1,219 @@
+# Inner product spaces 
+
+## Definition
+
+An introduction of length in a vector space may be formulated in terms of an inner product space. 
+
+> *Definition 1*: an **inner product** $V$ is an operation on $V$ that assigns, to each pair of vectors $\mathbf{x},\mathbf{y} \in V$ a real number $\langle \mathbf{x},\mathbf{y}\rangle$ satisfying the following conditions
+>
+> 1. $\langle \mathbf{x},\mathbf{x}\rangle > 0, \text{ for } \mathbf{x} \in V\backslash\{\mathbf{0}\} \text{ and }  \langle \mathbf{x},\mathbf{x}\rangle = 0, \; \text{for } \mathbf{x} = \mathbf{0}$,
+> 2. $\langle \mathbf{x},\mathbf{y}\rangle = \overline{\langle \mathbf{y},\mathbf{x}\rangle}, \; \forall \mathbf{x}, \mathbf{y} \in V$, 
+> 3. $\langle a \mathbf{x} + b \mathbf{y}, \mathbf{z}\rangle = a \langle \mathbf{x},\mathbf{z}\rangle + b \langle \mathbf{y},\mathbf{z}\rangle, \; \forall \mathbf{x}, \mathbf{y}, \mathbf{z} \in V \text{ and } a,b \in \mathbb{K}$.
+
+A vector space $V$ with an inner product is called an **inner product space**. 
+
+### Euclidean inner product spaces
+
+The standard inner product on the Euclidean vector spaces $V = \mathbb{R}^n$ with $n \in \mathbb{N}$ is given by the scalar product defined by
+
+$$
+    \langle \mathbf{x},\mathbf{y}\rangle = \mathbf{x}^T \mathbf{y}, 
+$$
+
+for all $\mathbf{x},\mathbf{y} \in V$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+This can be extended to matrices $V = \mathbb{R}^{m \times n}$ with $m,n \in \mathbb{N}$ for which an inner product may be given by
+
+$$
+    \langle A, B\rangle = \sum_{i=1}^m \sum_{j=1}^n a_{ij} b_{ij}, 
+$$
+
+for all $A, B \in V$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+### Function inner product spaces
+
+Let $V$ be a function space with a domain $X$. An inner product on $V$ may be defined by
+
+$$
+    \langle f, g\rangle = \int_X \bar f(x) g(x) dx 
+$$
+
+for all $f,g \in V$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+### Polynomial inner product spaces
+
+Let $V$ be a polynomial space of degree $n \in \mathbb{N}$ with the set of numbers $\{x_i\}_{i=1}^n \subset \mathbb{K}^n$. An inner product on $V$ may be defined by
+
+$$
+    \langle p, q \rangle = \sum_{i=1}^n \bar p(x_i) q(x_i),
+$$
+
+for all $p,q \in V$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Properties of inner product spaces
+
+> *Definition 2*: let $V$ be an inner product space, the Euclidean length $\|\mathbf{v}\|$ of a vector $\mathbf{v}$ is defined as
+>
+> $$
+>   \|\mathbf{v}\| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle},
+> $$
+>
+> for all $\mathbf{v} \in V$. 
+
+Which is consistent with Euclidean geometry. According to definition 1 the distance between two vectors $\mathbf{v}, \mathbf{w} \in V$ is $\|\mathbf{v} - \mathbf{w}\|$.
+
+> *Definition 3*: let $V$ be an inner product space, the vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal if
+>
+> $$
+>   \langle \mathbf{u}, \mathbf{v} \rangle = 0,
+> $$
+>
+> for all $\mathbf{u}, \mathbf{v} \in V$. 
+
+A pair of orthogonal vectors will satisfy the theorem of Pythagoras.
+
+> *Theorem 1*: let $V$ be an inner product space and $\langle \mathbf{u}, \mathbf{v} \rangle = 0$ are orthogonal then
+>
+> $$
+>   \|\mathbf{u} + \mathbf{v}\|^2 = \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2,
+> $$
+>
+> for all $\mathbf{u}, \mathbf{v} \in V$. 
+
+??? note "*Proof*:"
+
+    let $V$ be an inner product space and let $\mathbf{u}, \mathbf{v} \in V$ be orthogonal, then
+
+    $$
+    \begin{align*}
+        \|\mathbf{u} + \mathbf{v}\|^2 &= \langle \mathbf{u} + \mathbf{v}, \mathbf{u} + \mathbf{v}\rangle, \\
+                                    &= \langle \mathbf{u}, \mathbf{u} \rangle + 2 \langle \mathbf{u}, \mathbf{v} \rangle + \langle \mathbf{v}, \mathbf{v} \rangle, \\
+                                    &= \|\mathbf{u}\|^2 + \|\mathbf{v}\|^2. 
+    \end{align*}
+    $$
+
+Interpreted in $\mathbb{R}^2$ this is just the familiar Pythagorean theorem.
+
+> *Definition 4*: let $V$ be an inner product space then the **scalar projection** $a$ of $\mathbf{u}$ onto $\mathbf{v}$ is defined as
+>
+> $$
+>   a = \frac{1}{\|\mathbf{v}\|} \langle \mathbf{u}, \mathbf{v} \rangle,
+> $$
+>
+> for all $\mathbf{u} \in V$ and $\mathbf{v} \in V \backslash \{\mathbf{0}\}$. 
+>
+> The **vector projection** $p$ of $\mathbf{u}$ onto $\mathbf{v}$ is defined as
+>
+> $$
+> \mathbf{p} = a \bigg(\frac{1}{\|\mathbf{v}\|} \mathbf{v}\bigg) = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\langle \mathbf{v}, \mathbf{v} \rangle} \mathbf{v},
+> $$
+>
+> for all $\mathbf{u} \in V$ and $\mathbf{v} \in V \backslash \{\mathbf{0}\}$. 
+
+It may be observed that $\mathbf{u} - \mathbf{p}$ and $\mathbf{p}$ are orthogonal since $\langle \mathbf{p}, \mathbf{p} \rangle = a^2$ and $\langle \mathbf{u}, \mathbf{p} \rangle = a^2$ which implies 
+
+$$
+    \langle \mathbf{u} - \mathbf{p}, \mathbf{p} \rangle = \langle \mathbf{u}, \mathbf{p} \rangle - \langle \mathbf{p}, \mathbf{p} \rangle = a^2 - a^2 = 0.
+$$
+
+Additionaly, it may be observed that $\mathbf{u} = \mathbf{p}$ if and only if $\mathbf{u}$ is a scalar multiple of $\mathbf{v}$; $\mathbf{u} = b \mathbf{v}$ for some $b \in \mathbb{K}$. Since 
+
+$$
+    \mathbf{p} = \frac{\langle b \mathbf{v}, \mathbf{v} \rangle}{\langle \mathbf{v}, \mathbf{v} \rangle} \mathbf{v} = b \mathbf{v} = \mathbf{u}.
+$$
+
+> *Theorem 2*: let $V$ be an inner product space then
+>
+> $$
+>   | \langle \mathbf{u}, \mathbf{v} \rangle | \leq \| \mathbf{u} \| \| \mathbf{v} \|,
+> $$
+>
+> is true for all $\mathbf{u}, \mathbf{v} \in V$. With equality only holding if and only if $\mathbf{u}$ and $\mathbf{v}$ are linearly dependent.
+
+??? note "*Proof*:"
+
+    let $V$ be an inner product space and let $\mathbf{u}, \mathbf{v} \in V$. If $\mathbf{v} = \mathbf{0}$, then
+
+    $$
+        | \langle \mathbf{u}, \mathbf{v} \rangle | = 0 = \| \mathbf{u} \| \| \mathbf{v} \|,
+    $$
+
+    If $\mathbf{v} \neq \mathbf{0}$, then let $\mathbf{p}$ be the vector projection of $\mathbf{u}$ onto $\mathbf{v}$. Since $\mathbf{p}$ is orthogonal to $\mathbf{u} - \mathbf{p}$ it follows that
+
+    $$
+        \| \mathbf{p} \|^2 + \| \mathbf{u} - \mathbf{p} \|^2 = \| \mathbf{u} \|^2,
+    $$
+
+    thus 
+
+    $$
+        \frac{1}{\|\mathbf{v}\|^2} \langle \mathbf{u}, \mathbf{v} \rangle^2 = \| \mathbf{p}\|^2 = \| \mathbf{u} \|^2 - \| \mathbf{u} - \mathbf{p} \|^2,
+    $$
+
+    and hence
+
+    $$
+        \langle \mathbf{u}, \mathbf{v} \rangle^2 = \|\mathbf{u}\|^2 \|\mathbf{v}\|^2 - \|\mathbf{u} - \mathbf{p}\|^2 \|\mathbf{v}\|^2 \leq \|\mathbf{u}\|^2 \|\mathbf{v}\|^2,
+    $$
+
+    therefore
+
+    $$
+        | \langle \mathbf{u}, \mathbf{v} \rangle | \leq \| \mathbf{u} \| \| \mathbf{v} \|.
+    $$
+
+    Equality holds if and only if $\mathbf{u} = \mathbf{p}$. From the above observations, this condition may be restated to linear dependence of $\mathbf{u}$ and $\mathbf{v}$. 
+
+A consequence of the Cauchy-Schwarz inequality is that if $\mathbf{u}$ and $\mathbf{v}$ aer nonzero vectors in an inner product space then
+
+$$
+    -1 \leq \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\| \|\mathbf{v}\|} \leq 1,
+$$
+
+and hence there is a unique angle $\theta \in [0, \pi]$ such that 
+
+$$
+    \cos \theta = \frac{\langle \mathbf{u}, \mathbf{v} \rangle}{\|\mathbf{u}\| \|\mathbf{v}\|}.
+$$
+
+## Normed spaces
+
+> *Definition 5*: a vector space $V$ is said to be a **normed linear space** if to each vector $\mathbf{v} \in V$ there is associated a real number $\| \mathbf{v} \|$ satisfying the following conditions
+>
+> 1. $\|\mathbf{v}\| > 0, \text{ for } \mathbf{v} \in V\backslash\{\mathbf{0}\} \text{ and } \| \mathbf{v} \| = 0, \text{ for } \mathbf{v} = \mathbf{0}$,
+> 2. $\|a \mathbf{v}\| = |a| \|\mathbf{v}\|, \; \forall \mathbf{v} \in V \text{ and } a \in \mathbb{K}$,
+> 3. $\| \mathbf{v} + \mathbf{w}\| \geq \|\mathbf{v}\| + \| \mathbf{w}\|, \; \forall \mathbf{v}, \mathbf{w} \in V$,
+>
+> is called the **norm** of $\mathbf{v}$. 
+
+With the third condition, the *triangle inequality*. 
+
+> *Theorem 3*: let $V$ be an inner product space then
+>
+> $$
+>   \| \mathbf{v} \| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle},
+> $$
+>
+> for all $\mathbf{v} \in V$ defines a norm on $V$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+We therefore have that the Euclidean length (definition 2) is a norm, justifying the notation.