I was able to derive the formula for the Ridge Regression Coefficient Estimate $\hat{\boldsymbol{\beta}}^{ridge}$. However, I am not 100% sure what it means in terms of showing that the Ridge Regression Coefficient Estimate is linear in Y, but my intuition is that if I expand the following term by using SVD method on matrix X which has centered columns (leading to a symmetric matrix X): $$ (\textbf{X}^{T}\textbf{X}+\lambda\textbf{I})^{-1}\textbf{X}^{T} = \textbf{V}(\textbf{D}^{2}+\lambda \textbf{I})^{-1}\textbf{V}^{T}\textbf{VD}\textbf{U}^{T} $$the resulting result will have a linear form like $a+bx$.
Am I correct?
If not, please give me some tips.
$\newcommand{\y}{\mathbf{y}}$The problem requires you to show that $\widehat{\boldsymbol{\beta}}^{ridge}$ is of the form $M\y$ for some matrix $M$.