Matrices & Vectors

Matrix
Matrix: rectangular array of numbers
Dimension of matrix: number of rows $\times$ number of columns
$A_{ij}$ : $i$ , $j$ entry in the $i^{th}$ row, $j^{th}$ column

e.g.

$\begin{bmatrix} a & b & c \newline d & e & f \newline g & h & i \newline j & k & l \end{bmatrix}$

dimension: $4\times3$ or $\mathbb{R^{4\times3}}$
$A_{11}=a$
$A_{32}=h$

Vector
Vector: $n\times1$ matrix
$v_{i}$ : $i^{th}$ element

e.g.

$\begin{bmatrix} a \newline b \newline c \end{bmatrix}$

dimension: 3-dimensional vector or $\mathbb{R^{3}}$
$v_{1}=a$
$v_{3}=c$

1-indexed vector:

$\begin{bmatrix} y_1 \newline y_2 \newline y_3 \end{bmatrix}$

0-indexed vector:

$\begin{bmatrix} y_0 \newline y_1 \newline y_2 \end{bmatrix}$