Updated linear algebra section.

config/en/mkdocs.yaml

@@ -81,7 +81,9 @@ nav:
       - 'Complex numbers': mathematics/number-theory/complex-numbers.md
     - 'Linear algebra':
       - 'Systems of linear equations': mathematics/linear-algebra/systems-of-linear-equations.md
-      - 'Matrices': mathematics/linear-algebra/matrices.md
+      - 'Matrices': 
+        - 'Matrix arithmetic': mathematics/linear-algebra/matrices/matrix-arithmetic.md
+        - 'Matrix algebra': mathematics/linear-algebra/matrices/matrix-algebra.md
     - 'Calculus': 
       - 'Limits': mathematics/calculus/limits.md
       - 'Continuity': mathematics/calculus/continuity.md

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

@@ -1 +0,0 @@
-# Matrices

docs/en/mathematics/linear-algebra/matrices/matrix-algebra.md

@@ -0,0 +1 @@
+# Matrix algebra

docs/en/mathematics/linear-algebra/matrices/matrix-arithmetic.md

@@ -0,0 +1,91 @@
+# Matrix arithmetic
+
+## Definitions
+
+> *Definition*: let $A$ be a $m \times n$ *matrix* given by
+>
+> $$
+> A = \begin{pmatrix} a_{11} & a_{12}& \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}
+> $$
+>
+> with $a_{ij}$ referred to as the entries of $A$ or scalars in general, with $(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}$. For real entries in $A$ we may denote $A \in \mathbb{R}^{m \times n}$.
+
+This matrix may be denoted in a shorter way by $A = (a_{ij})$. 
+
+> *Definition*: let $\mathbf{x}$ be a $1 \times n$ matrix, referred to as *row vector* given by
+>
+> $$
+>   \mathbf{x} = \begin{pmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{pmatrix}
+> $$
+>
+> with $x_i$ referred to as the entries of $\mathbf{x}$, with $i \in \{1, \dots, n\}$. For real entries we may denote $\mathbf{x} \in \mathbb{R}^n$.
+
+<br>
+
+> *Definition*: let $\mathbf{x}$ be a $n \times 1$ matrix, referred to as *column vector* given by
+>
+> $$
+>   \mathbf{x} = (x_1, x_2, \dots, x_n)
+> $$
+>
+> with $x_i$ referred to as the entries of $\mathbf{x}$, with $i \in \{1, \dots, n\}$. Also for the column vector we have for real entries $\mathbf{x} \in \mathbb{R}^n$. 
+
+From these two definitions it may be observed that row and column vectors may be used interchangebly, however using both it is important to state the difference. Best practice is to always work with row vectors and take the transpose if necessary.
+
+## Matrix operations
+
+> *Definition*: two $m \times n$ matrices $A$ and $B$ are said to be **equal** if $a_{ij} = b_{ij}$ for each $i(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}$.
+
+<br>
+
+> *Definition*: if $A$ is an $m \times n$ matrix and $\alpha$ is a scalar, then $\alpha A$ is the $m \times n$ matrix whose $(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}$ entry is $\alpha a_{ij}$. 
+
+<br>
+
+> *Definition*: if $A = (a_{ij})$ and $B = (b_{ij})$ are both $m \times n$ matrices, then the sum $A + B$ is the $m \times n$ matrix whose $(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}$ entry is $a_{ij} + b_{ij}$ for each ordered pair $(i,j)$.
+
+If $A$ is an $m \times n$ matrix and $\mathbf{x}$ is a vector in $\mathbb{R}^n$, then
+
+$$
+    A \mathbf{x} = x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \dots + x_n \mathbf{a}_n
+$$
+
+with $A = (\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n})$. 
+
+> *Definition*: if $\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n}$ are vectors in $\mathbb{R}^m$ and $x_1, x_2 \dots, x_n$ are scalars, then a sum of the form
+>
+> $$
+>   x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \dots + x_n \mathbf{a}_n
+> $$
+>
+> is said to be a **linear combination** of the vectors $\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n}$. 
+
+<br>
+
+> *Theorem*: a linear system $A \mathbf{x} = \mathbf{b}$ is consistent if and only if $\mathbf{b}$ can be written as a linear combination of the column vectors $A$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Transpose matrix
+
+> *Definition*: the transpose of an $m \times n$ matrix A is the $n \times m$ matrix $B$ defined by
+>
+> $$
+>   b_{ji} = a_{ij}
+> $$
+>
+> for $j \in \{1, \dots, n\}$ and $i \in \{1, \dots m\}$. The transpose of $A$ is denoted by $A^T$.
+
+<br>
+
+> *Definition*: an $n \times n$ matrix $A$ is said to be **symmetric** if $A^T = A$.
+
+## Matrix multiplication
+
+> *Definition*: if $A = (a_{ij})$ is an $m \times n$ matrix and $B = (b_{ij})$ is an $n \times r$ matrix, then the product $A B = C = (c_{ij})$ is the $m \times r$ matrix whose entries are defined by
+>
+> $$
+>   c_{ij} = \mathbf{a}_i \mathbf{b}_j = \sum_{k=1}^n a_{ik} b_{kj}
+> $$