In order to understand the relation among the predictors \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), we condiser a simple polynomial fit. We assume our data can represented by a fourth-order polynomial. For the $i$th component we have
$$ \tilde{y}_i = \beta_0+\beta_1x_i+\beta_2x_i^2+\beta_3x_i^3+\beta_4x_i^4. $$we have five predictors/features. The first is the intercept \( \beta_0 \). The other terms are \( \beta_i \) with \( i=1,2,3,4 \). Furthermore we have \( n \) entries for each predictor. It means that our design matrix is an \( n\times p \) matrix \( \boldsymbol{X} \).