Are the following statements true of false? I've tried to prove them, but don't to get anywhere.
I) If $\langle l,m \rangle = -\langle m,l \rangle$ in a vector space with inner product, then $l \perp m$.
II) If $\langle l,m \rangle = -\langle l,n \rangle$ in a vector space with inner product, then $l\perp(m-n)$.
That's probably a very noob question, but I just can't seem to get it right.
I) $\langle l,m \rangle = -\langle m,l \rangle = - \overline{\langle l,m \rangle}$ so if $\langle l,m \rangle = a+bi$, then $$a+bi = -(a-bi) = -a+bi \Rightarrow a = 0.$$ In particular, $\langle l,m \rangle$ just has to be imaginary not necessarily $0$.
II) In order for $l \perp (m-n)$ to be true, we need $$0 = \langle l, m-n \rangle = \langle l,m \rangle - \langle l,n \rangle = -\langle n,l \rangle - \langle l,n \rangle.$$ Try to cook up an example of $l$ and $n$ where this is not true.