$||y-Xb-e_{1}c||^{2}$ = $(y_{1}-w_{1}^{T}b-c)^{2}$ $+$ $||Y-Wb||^{2}$,
Where $y_{1}$ is the first observation in $y, y=\left(\begin{array}{c} y_{1} \\ Y \end{array}\right), X = \left(\begin{array}{c} w_{1}^{T} \\ W \end{array}\right), and W = \left(\begin{array}{c} w_{2}^{T} \\ ... w_{n}^{T}\end{array}\right)$
$c$ is a constant, and $e_{1}$ is the vector $(1,0,...,0)$, so $e_{1}c = (c,0,...,0)$
Why is the case that $\hat{c}$ takes the same value for all choices of $\hat{B}$ that minimizes $||Y-Wb||^{2}$ if and only if $w_{1}$ lies in $\mathcal{W}$, the subspace spanned by ${w_{i} : 2 \leq i \leq n}$?
Does this mean that $\hat{c}$ is zero?