TITLE: Taking the Inner Product with the Zero Vector
QUESTION [6 upvotes]: Is it possible for the inner product of any vector with the zero vector $ \mathbf{0} $ to be nonzero? Or must it always be zero? I'm struggling to find a counterexample. That is, is the following statement correct?
$$ \langle \mathbf{v}, \mathbf{0}\rangle = 0 \space \forall \space v \in V$$

REPLY [5 votes]: Think of it simply like this.
$\langle \mathbf{v} , \mathbf0 \rangle = \langle \mathbf{v}, \mathbf{0}*\mathbf{0} \rangle$
Here let one of the zeroes be the scalar,
then the RHS will look like
$\langle \mathbf{v} , \mathbf0 \rangle = \langle \mathbf{v} , \mathbf{0}*\mathbf{0} \rangle = 0\cdot\langle \mathbf{v} , \mathbf{0} \rangle$
which is $0$.