The Line That Misses Every Point

You are fitting the straight line $\hat{y} = \beta_0 + \beta_1 t$ to three data points by least squares:

In matrix form this is $y \approx X\beta$ with

$X = \begin{pmatrix}1 & 0\\ 1 & 1\\ 1 & 2\end{pmatrix}, \qquad y = \begin{pmatrix}6\\ 0\\ 0\end{pmatrix}, \qquad \beta = \begin{pmatrix}\beta_0\\ \beta_1\end{pmatrix}$

The three points are not collinear, so no exact solution exists — least squares projects $y$ onto the column space of $X$ instead.

Find the least-squares coefficients $\hat\beta$ and the residual sum of squares $\text{RSS} = \|y - X\hat\beta\|_2^2$.

(Sanity check available to you: the residual vector must come out orthogonal to both columns of $X$.)