I have a homework problem that seems wrong. Previous to the problem we showed that the inner product of a vector space $V$ on the field $\mathbb{C}$ defined as $$ \langle v,w \rangle = a^i(b^j)^*\delta_{ij} $$ where $a^i, b^j$ are the components of the vectors $v = a^i e_{i}$, $w = b^i e_{i}$ satisfy the inner product space axioms. Namely \begin{align*} &\cdot \langle v,w \rangle = \langle w,v \rangle^* \\ &\cdot \langle au + bv,w \rangle = a\langle u,v \rangle + b\langle v,w \rangle \\ &\cdot \forall v\in \mathbb{C}\backslash\{0\}, \quad \langle v,v \rangle > 0 \\ &\cdot \langle 0,0 \rangle = 0 \end{align*} The problem asks us to consider the inner product defined as: $$ \langle a^i e_i, b^j e_j \rangle = b^i(a^j)^* \delta_{ij} $$ and explain why this does not define an inner product vector space.
It seems to me that it satisfies all the necessary axioms. Where am I wrong?