The multiple linear regression model is $\mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}$ where $\mathbf{Y} = (Y_1, \ldots, Y_n)^T$ is the $n \times 1$ response vector, $\mathbf{X}$ is the $n \times p$ design matrix (with rows $(1, x_{i1}, \ldots, x_{i,p-1})$), $\boldsymbol{\beta} = (\beta_0, \beta_1, \ldots, \beta_{p-1})^T$ is the $p \times 1$ parameter vector, and $\boldsymbol{\epsilon} \sim N(\mathbf{0}, \sigma^2 \mathbf{I}_n)$.

The OLS estimator minimizes $\|\mathbf{Y} - \mathbf{X}\boldsymbol{\beta}\|^2$ and is given by $\hat{\boldsymbol{\beta}} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y}$ provided $\mathbf{X}^T\mathbf{X}$ is invertible (i.e., $\mathbf{X}$ has full column rank $p$). The fitted values are $\hat{\mathbf{Y}} = \mathbf{X}\hat{\boldsymbol{\beta}} = \mathbf{H}\mathbf{Y}$, where $\mathbf{H} = \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T$ is the hat matrix.