1. (3 points) (Orthonormal Bases/Gram-Schmidt) Consider the matrix
$A = \begin{bmatrix} \vec{a}_1 & \vec{a}_3 & \vec{a}_1 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 4 \\ 0 & 0 & 5 \\ 0 & 3 & 6 \end{bmatrix}$.
(a) Find an orthonormal basis \(\{\vec{q}_1, \vec{q}_2, \vec{q}_3\}\) for the columns of the matrix A.
(b) Find the matrix R for which A = QR, i.e., the matrix which expresses the vectors
$\vec{a}_1, \vec{a}_2$ and $\vec{a}_3$ as linear combinations of the orthonormal basis vectors. (Hint: what
is the inverse of Q, as discussed in class?).
(c) From the result above, find the constants $c_{11}, c_{12}$ and $c_{13}$ for which
$\vec{a}_1 = c_{11}\vec{q}_1 + c_{12}\vec{q}_2 + c_{13}\vec{q}_3.$
