1.1.6 The Linear Transformation

We define $M$ to be a square $n \times n$ matrix, and $T(v)$ de­fines the linear trans­for­ma­tion $M$ per­forms on the $1 \times n$ column vector $v$

$T(N)$ de­fines the left-handed linear trans­for­ma­tion $M$ per­forms on an­oth­er $n \times n$ matrix $N$

We can have $v \in \mathbb{R}^{n}$ or $\in \mathbb{C}^{n}$, if the linear trans­forms hold to $T:\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ and $T:\mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$.