Basic terminology and notation

**Basic terminology and notation** for matrices establishes the foundational language used throughout linear algebra.

A matrix is denoted by a capital letter (e.g., $A$), while its entries are denoted by the corresponding lowercase letter with subscripts (e.g., $a_{ij}$ or $A_{ij}$), where $i$ indicates the row and $j$ indicates the column.

Key notation:
- $A \in \mathbb{R}^{m \times n}$ means $A$ is an $m \times n$ matrix with real entries
- $a_{ij}$ denotes the entry in row $i$, column $j$
- $A^T$ denotes the transpose of $A$
- $A^{-1}$ denotes the inverse of $A$ (when it exists)
- $I$ or $I_n$ denotes the $n \times n$ identity matrix
- $0$ or $0_{m \times n}$ denotes the zero matrixCopy