(a) The two sets of basis vectors $\mathbf{e}_{i}$ and $\mathbf{e}_{i}^{\prime}$ (where $i=1,2,3$ ) are related by

$$\mathbf{e}_{i}^{\prime}=R_{i j} \mathbf{e}_{j}$$

where $R_{i j}$ are the entries of a rotation matrix. The components of a vector $\mathbf{v}$ with respect to the two bases are given by

$$\mathbf{v}=v_{i} \mathbf{e}_{i}=v_{i}^{\prime} \mathbf{e}_{i}^{\prime}$$

Derive the relationship between $v_{i}$ and $v_{i}^{\prime}$.

(b) Let $\mathbf{T}$ be a $3 \times 3$ array defined in each (right-handed orthonormal) basis. Using part (a), state and prove the quotient theorem as applied to $\mathbf{T}$.