Three Columns, Two Directions

A model is fitted to four training examples with three features and no intercept term. The design matrix $A$ holds one example per row, and $\mathbf{y}$ holds the targets:

Least squares picks the weight vector $\hat{\boldsymbol{\beta}}$ that makes the predictions $A\hat{\boldsymbol{\beta}}$ as close to $\mathbf{y}$ as possible:

$\hat{\boldsymbol{\beta}} = \arg\min_{\boldsymbol{\beta}} \;\lVert \mathbf{y} - A\boldsymbol{\beta} \rVert$

Geometrically, $A\hat{\boldsymbol{\beta}}$ is the orthogonal projection of $\mathbf{y}$ onto the column space of $A$ — the set of all vectors the model is capable of producing. Whatever is left over is the error no choice of weights can remove.

The columns of $A$ are not mutually orthogonal, so you will have to orthogonalise them before projecting. Work carefully: check what each column actually contributes before you use it.

What is $\lVert \mathbf{y} - A\hat{\boldsymbol{\beta}} \rVert$, the distance from $\mathbf{y}$ to the column space of $A$? Round your answer to 2 decimal places.