Given a non-zero vector $v_{i}$, any $3 \times 3$ symmetric matrix $T_{i j}$ can be expressed as

$$T_{i j}=A \delta_{i j}+B v_{i} v_{j}+\left(C_{i} v_{j}+C_{j} v_{i}\right)+D_{i j}$$

for some numbers $A$ and $B$, some vector $C_{i}$ and a symmetric matrix $D_{i j}$, where

$$C_{i} v_{i}=0, \quad D_{i i}=0, \quad D_{i j} v_{j}=0,$$

and the summation convention is implicit.

Show that the above statement is true by finding $A, B, C_{i}$ and $D_{i j}$ explicitly in terms of $T_{i j}$ and $v_{j}$, or otherwise. Explain why $A, B, C_{i}$ and $D_{i j}$ together provide a space of the correct dimension to parameterise an arbitrary symmetric $3 \times 3$ matrix $T_{i j}$.