A function $L: \mathbb{R}^n \to \mathbb{R}^m$ is called a linear map if it respects addition and scalar multiplication. Symbolically, for a map to be linear, we must have that

$$\begin{aligned} L(\vec{v}+\vec{w}) &= L(\vec{v})+L(\vec{w}) & \text{for all }\vec{v},\vec{w} \in \mathbb{R}^n \\ \end{aligned}$$

and also

$$\begin{aligned} L(a\vec{v}) &= a L(\vec{v}) \quad & \text{for all }a \in \mathbb{R} \text{ and } \vec{v} \in \mathbb{R}^n \\ \end{aligned}$$

To each linear map $L: \mathbb{R}^n \to \mathbb{ℝ}^m$ we associate a $m \times n$ matrix $A_L$ called the matrix of the linear map with respect to the standard coordinates. It is defined by setting $a_{i,j}$ to be the $i^{th}$ component of $L(e_j)$. In other words, the $j^{th}$ column of the matrix $A_L$ is the vector $L(e_j)$.

Source: m2o2c2.

# posted by rot13(Unafba Pune) @ 8:47 AM