When predicting labels for an entire dataset consisting of $n$ examples, it is mathematically convenient to organize the features into a design matrix $\mathbf{X} \in \mathbb{R}^{n imes d}$, where each row represents an example and each column a feature. The vector of predictions $\hat{\mathbf{y}} \in \mathbb{R}^n$ for all examples can then be concisely computed using a matrix-vector product with the weight vector $\mathbf{w}$, adding the bias term $b$ via broadcasting: $\hat{\mathbf{y}} = \mathbf{X} \mathbf{w} + b$.