Four Weights, Two Real Directions

A linear model predicts with

$\hat{y} = w_1 f_1 + w_2 f_2 + w_3 f_3 + w_4 f_4$

where each $f_j$ is the column of feature $j$'s values across the whole dataset — so each $f_j$ is a vector in $\mathbb{R}^3$, and $\hat{y}$ is the vector of the model's three predictions.

The dataset has three rows and four features:

Training returned the weight vector

$w = \begin{pmatrix} 2 \\ 3 \\ 4 \\ -1 \end{pmatrix}$

The four feature columns are not independent: they span only a $2$-dimensional subspace of $\mathbb{R}^3$, and $\{f_1,\ f_2\}$ is a basis of that subspace. Because of this, the trained weights are not the only ones that produce these predictions — there is a unique pair $w_1', w_2'$ with

$w_1 f_1 + w_2 f_2 + w_3 f_3 + w_4 f_4 \;=\; w_1' f_1 + w_2' f_2$

holding as an equality of vectors, so the two-feature model reproduces the four-feature model's prediction on every row.

Report the length of that reduced weight vector,

$\left\lVert \begin{pmatrix} w_1' \\ w_2' \end{pmatrix} \right\rVert = \sqrt{(w_1')^2 + (w_2')^2}$

rounded to 2 decimal places.