From 4e6cd51169b71458e01cf3b99f14eea68f65b8b9 Mon Sep 17 00:00:00 2001 From: Luc Date: Sat, 6 Jan 2024 21:22:26 +0100 Subject: [PATCH] Updated linear algebra section. --- config/en/mkdocs.yaml | 4 +- .../en/mathematics/linear-algebra/matrices.md | 1 - .../linear-algebra/matrices/matrix-algebra.md | 1 + .../matrices/matrix-arithmetic.md | 91 +++++++++++++++++++ 4 files changed, 95 insertions(+), 2 deletions(-) delete mode 100644 docs/en/mathematics/linear-algebra/matrices.md create mode 100644 docs/en/mathematics/linear-algebra/matrices/matrix-algebra.md create mode 100644 docs/en/mathematics/linear-algebra/matrices/matrix-arithmetic.md diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index d03c4d3..d9e682d 100755 --- a/config/en/mkdocs.yaml +++ b/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 diff --git a/docs/en/mathematics/linear-algebra/matrices.md b/docs/en/mathematics/linear-algebra/matrices.md deleted file mode 100644 index ab19227..0000000 --- a/docs/en/mathematics/linear-algebra/matrices.md +++ /dev/null @@ -1 +0,0 @@ -# Matrices \ No newline at end of file diff --git a/docs/en/mathematics/linear-algebra/matrices/matrix-algebra.md b/docs/en/mathematics/linear-algebra/matrices/matrix-algebra.md new file mode 100644 index 0000000..2405241 --- /dev/null +++ b/docs/en/mathematics/linear-algebra/matrices/matrix-algebra.md @@ -0,0 +1 @@ +# Matrix algebra \ No newline at end of file diff --git a/docs/en/mathematics/linear-algebra/matrices/matrix-arithmetic.md b/docs/en/mathematics/linear-algebra/matrices/matrix-arithmetic.md new file mode 100644 index 0000000..0d76ff3 --- /dev/null +++ b/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 $\mathbf{x} \in \mathbb{R}^n$. + +
+ +> *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 $i(i,j) \in \{1, \dots, m\} \times \{1, \dots, n\}$. + +
+ +> *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 $\alpha a_{ij}$. + +
+ +> *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 $\mathbf{a_1}, \mathbf{a_2}, \dots, \mathbf{a_n}$. + +
+ +> *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 $A^T$. + +
+ +> *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 +> +> $$ +> c_{ij} = \mathbf{a}_i \mathbf{b}_j = \sum_{k=1}^n a_{ik} b_{kj} +> $$ \ No newline at end of file