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