I just had some confusion about linearity of inner product.
In my course notes, it's said that inner product is linear in first slot i.e.
$\langle u+w,v\rangle = \langle u,v\rangle + \langle w,v\rangle$ and $\langle au,v\rangle = a\langle u,v\rangle$
But when I searched for logical explanations, I stumbled upon a video which said that inner product is linear in second argument i.e.
$\langle u,v+w\rangle = \langle u,v\rangle + \langle u,w\rangle$ and $\langle u,\lambda v\rangle = \lambda \langle u,v\rangle$
But neither of the sources said about the other slot/argument.
So, in which slot/argument is inner product actually linear?