1. [10 marks] Find the function of the form $y = \beta_0 + \beta_1 x + \beta_2 x^2$ which is closest (in least-squares sense) to going through the points: (0,4), (1, 2), (2, 3), (3, 6).
2. [16 marks] The matrix A below has eigenvalues 10 and 4.
$A = \begin{bmatrix} 10 & -1 & 0 & 0 \\ -2 & 9 & 0 & -1 \\ 0 & 0 & 4 & 0 \\ 2 & 3 & 0 & 11 \end{bmatrix}$
(a) Find the dimensions of the subspaces $ker((A - \lambda I)^k)$ (the nullspace of $(A - \lambda I)^k$) for $\lambda = 10$ and 4 and $k = 1, 2, 3, ...$ and using that information, find the Jordan Canonical Form of A.
(b) Find a basis $\beta$ (which must contain the vector $\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$ so that $[A]_\beta$ is the Jordan Canonical Form of A.
