If our design matrix \( \hat{X} \) which enters the linear regression problem $$ \begin{align} \hat{\beta} & = (\hat{X}^{T} \hat{X})^{-1} \hat{X}^{T} \hat{y}, \tag{1} \end{align} $$ has linearly dependent column vectors, we will not be able to compute the inverse of \( \hat{X}^T\hat{X} \) and we cannot find the parameters (estimators) \( \beta_i \). The estimators are only well-defined if \( (\hat{X}^{T}\hat{X})^{-1} \) exits. This is more likely to happen when the matrix \( \hat{X} \) is high-dimensional. In this case it is likely to encounter a situation where the regression parameters \( \beta_i \) cannot be estimated.
The ad hoc approach which was introduced in the 70s was simply to add a diagonal component to the matrix to invert, that is we change $$ \hat{X}^{T} \hat{X} \rightarrow \hat{X}^{T} \hat{X}+\lambda \hat{I}, $$ where \( \hat{I} \) is the identity matrix.