## Matrices and Vectors

Matrices are 2-dimensional arrays:

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

The above matrix has four rows and three columns, so it is a 4 x 3 matrix.

A vector is a matrix with one column and many rows:

$\begin{bmatrix} w \\ x \\ y \\ z \end{bmatrix}$

So vectors are a subset of matrices. The above vector is a 4 x 1 matrix.

#### Notation and terms:

- $A_{ij}$ refers to the element in the $i$th row and $j$th column of matrix $A$.
- $A$ vector with 'n' rows is referred to as an 'n'-dimensional vector.
- $v_i$ refers to the element in the $i$th row of the vector.
- In general, all our vectors and matrices will be 1-indexed. Note that for some programming languages, the arrays are 0-indexed.
- Matrices are usually denoted by uppercase names while vectors are lowercase.
- "Scalar" means that an object is a single value, not a vector or matrix.
- $\mathbb{R}$ refers to the set of scalar real numbers.
- $\mathbb{R^n}$ refers to the set of n-dimensional vectors of real numbers.
- $\mathbb{R^{m \times n}}$ refers to the set of $m \times n$ matrices of real numbers.