Inner product spaces and linear maps

Question

Let $V$ and $W$ be inner product spaces over $K$ and let $T,S:V \rightarrow W$ be linear maps. Show the following:

• If $K=\mathbb R$, then $\langle T(v_1),T(v_2) \rangle_W = 1/4 \langle T(v_1+v_2),T(v_1+v_2) \rangle_W - 1/4 \langle T(v_1-v_2),T(v_1-v_2) \rangle_W$.

• If $K=\mathbb C$, then $\langle T(v_1),S(v_2) \rangle_W = 1/4 \sum\limits_{k=1}^4 i^k \langle T(v_1+i^kv_2),S(v_1+i^kv_2) \rangle_W$ (where $i^2=-1$).

Answer 1

Hint: Expand the right sides and see that everything simplifies, giving you the left sides.

Answer 2

For the real case, we have $1/4\langle T(v_1+v_2),T(v_1+v_2)\rangle-1/4\langle T(v_1-v_2),T(v_1-v_2)\rangle$

$=1/4\langle T(v_1)+T(v_2),T(v_1)+T(v_2)\rangle-1/4\langle T(v_1)-T(v_2),T(v_1)-T(v_2)\rangle$

$=1/4\langle T(v_1),T(v_1)\rangle+1/4\langle T(v_1),T(v_2)\rangle+1/4\langle T(v_2),T(v_1)\rangle+1/4\langle T(v_2),T(v_2)\rangle$

$-1/4\langle T(v_1),T(v_1)\rangle+1/4\langle T(v_1),T(v_2)\rangle+1/4\langle T(v_2),T(v_1)\rangle-1/4\langle T(v_2),T(v_2)\rangle$

$=1/2\langle T(v_1),T(v_2)\rangle+1/2\langle T(v_2),T(v_1)\rangle=\langle T(v_1),T(v_2)\rangle$