Vector Function

Definition

Let $\mathbf{f}:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be a function that maps a vector $\mathbf{x}\in {\mathbb{R}}^n$ to a vector $\mathbf{y}\in {\mathbb{R}}^m$.

Then $\mathbf{f}$ is called a vector function and can be written as:

$$\mathbf{f}(\mathbf{x})=\left[\begin{array}{c}{f}_1(\mathbf{x})\\ f_2(\mathbf{x})\\ ⋮\\ f_m(\mathbf{x})\end{array}\right]$$

where each $f_i:{\mathbb{R}}^n\to \mathbb{R}$ is a scalar function.

Example

Let $\mathbf{f}:{\mathbb{R}}^2\to {\mathbb{R}}^3$ be defined as:

$$\mathbf{f}\left(\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\right)=\left[\begin{array}{c}{x}_1^2+x_2\\ 2x_1{x}_2\\ x_2^2-x_1\end{array}\right]$$

Here, $f_1(x_1,x_2)=x_1^2+x_2$, $f_2(x_1,x_2)=2x_1{x}_2$, $f_3(x_1,x_2)=x_2^2-x_1$.

For $\mathbf{x}=\left[\begin{array}{c}1\\ 2\end{array}\right]$:

$$\mathbf{f}\left(\left[\begin{array}{c}1\\ 2\end{array}\right]\right)=\left[\begin{array}{c}1+2\\ 2(1)(2)\\ 4-1\end{array}\right]=\left[\begin{array}{c}3\\ 4\\ 3\end{array}\right]$$