1. Vectors and Matrices

Linear Combination

The set $\mathbb{R}^{n}$ or an infinite line can be represented as linear combinations of vectors

Example 1 : $c\begin{bmatrix}1 \\ 0\end{bmatrix} + d\begin{bmatrix}0 \\ 1\end{bmatrix}$ spans $\mathbb{R}^2$

Example 2 : $c\begin{bmatrix}1 \\ 1\end{bmatrix} + d\begin{bmatrix}2 \\ 2\end{bmatrix}$ is an infinite line

 

 

Lengths and Dot (Inner) Products

Length : $\|v\| = \sqrt{v \cdot v} = \sqrt{ \sum\limits_{i=1}^{n} v_{i}^2}$, called the $l_{2}$-norm

Triangle inequality : $\|v+w\| = \|v\| + \|w\|$

 

Angle between vectors $v$ and $w$ is defined as the value $\theta$ such that

$cos(\theta)=\cfrac{v \cdot w} {\|v\| \|w\|}$

 

 

Linear Equations

A function $f$ is called linear if

$f(a\mathbf{x}+b\mathbf{y})=af(\mathbf{x})+bf(\mathbf{y})$

$f(x) = ax + b$ is not a linear function

 

Linear equation : equation expressed as "linear function = c"

Example : $x_{1} + 3x_{2} + 4x_{3} = 2$