I have a question in linear algebra 2, in the topic of inner product spaces:
$\text{Let V be an inner space product over field F (could be either}\>\mathbb{C}\>\text{or}\>\mathbb{R}\text{)}.\\ \text{Prove:}\>\forall u,v_1,\dots,v_m\in V,\forall \eta_1,\dots,\eta_m\in F:\left< u,\sum_{j=1}^{m}\eta_jv_j \right>=\sum_{j=1}^{m}\overline{\eta_j}\left<u,v_j\right>$
I started with this:
$\begin{array}{cccc} \left\langle u,\sum_{j=1}^{m}\eta_{j}v_{j}\right\rangle & = & \overline{\left\langle \sum_{j=1}^{m}\eta_{j}v_{j},u\right\rangle } & \because\text{hermiticity}\ \left\langle \ ,\ \right\rangle \\ & = & \left\langle \sum_{j=1}^{m}\overline{\eta_{j}v_{j}},\overline{u}\right\rangle & \because\text{properties of}\ \cdot_{\mathbb{C}},+_{\mathbb{C}}\\ \end{array}$
I also thought of playing with things (properties of $<\>,\>>$, and properties of operations in $\mathbb{C}$), and I just didn't get a direction.
I'd appreciate a hint/a direction a lot.
Observe \begin{align} \langle u, \eta_1v_1+\eta_2v_2\rangle =& \overline{\langle \eta_1v_1, u\rangle}+ \overline{\langle \eta_2v_2, u\rangle}\\ =& \overline{\eta_1 \langle v_1, u\rangle}+ \overline{\eta_2 \langle v_2, u\rangle}\\ =&\ \bar \eta_1\langle u, v_1\rangle +\bar\eta_2\langle u, v_2\rangle. \end{align}