An explicit feature mapping is a mathematical transformation of the form $f : \mathbb{R}^n \to \mathbb{R}^m$. When using explicit feature mappings, we compute $f(\vec{x})$ for each data point $\vec{x}$ in our feature space to project it into a new, typically higher-dimensional space where the data may become linearly separable. For example, if we are given the feature space $\vec{x} = [x_1, x_2]^T \in \mathbb{R}^2$, and we assume that our data is not linearly separable but is quadratically separable, we might define a feature mapping to be $f: \mathbb{R}^2 \to \mathbb{R}^3$ where $f([x_1, x_2]^T) = [x_1^2, x_2^2, 1]^T$.