(1) Let V be a three dimensional vector space with inner product $\langle .,.\rangle$, and let $\vec{b_1}, \vec{b_2}, \vec{b_3}$ be an orthonormal basis of V, i.e. each of them has length/norm equal to one and they are mutually perpendicular. Suppose further that there is linear operator $L: V \to V$ such that $L(\vec{b_1}) = 3\vec{b_2}$, $L(\vec{b_2}) = 7\vec{b_1}$, $L(\vec{b_3}) = -11\vec{b_3}$.
(a) For any given numbers x, y, z evaluate $L(x\vec{b_1} + y\vec{b_2} + z\vec{b_3})$.
[So that your answer does not involve L anymore.]
(b) Any vector $\vec{v} \in V$ can be written as $\vec{v} = x\vec{b_1} + y\vec{b_2} + z\vec{b_3}$ for some appropriate choice for the numbers x, y, z. Write down a formula that allows you to compute/express x, y, z in terms of $\vec{v}, \vec{b_1}, \vec{b_2}, \vec{b_3}$.
(c) Expressing both $\vec{v}$ as $\vec{v} = x\vec{b_1} + y\vec{b_2} + z\vec{b_3}$ and $L(\vec{v})$ as $L(\vec{v}) = \alpha\vec{b_1} + \beta\vec{b_2} + \gamma\vec{b_3}$ Find a 3 x 3 matrix M such that $\begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} = M \begin{pmatrix} x \\ y \\ z \end{pmatrix}$.
