I was given: $S$, a subspace of $\Bbb R^n$, where $\dim S = m \le n$; A basis $\{v_1, ... ,v_m\}$ of S; $x = x_s + x_s^p$, where $x_s$ is the orthogonal projection of x on $S$ and $x_s^p$ is orthogonal to $x_s$.
I was asked to show that the coeficients from the least squares solution equal the ones from the solution of the equation $V^T x = (V^T V)β$, which I managed to prove.
Now, in addition, I should prove that this equation still holds when I add one dimension to the basis of $S$.
Specifically, the prove must verify that the linear function associated whit the $m + 1$ case is bijective.
The problem: I have no cue on how to start...
Any thoughts would be very appreciated.