Im currently looking at inner products and was wondering why the inner product of any vector with the zero vector is equal to 0. I have researched on this and only found the information that the zero vector is orthogonal to all vectors but no proof alongside.
And hence I was wondering if anyone had any proof as to why this happens.
Thanks in advance.
Edit:
Thanks every for their comment, I was now wondering if anyone could help explain to me why the inner product of $\left \langle M \vec u,\vec v \right \rangle$ = $\left \langle \vec u,M^{T} \vec v \right \rangle$
where M is and $n*n$ square matrix.
Best Answer
Let us denote the inner product by $( \cdot|\cdot).$
Then
$$(0|x)=(0x|x)=0(x|x)=0$$
for all $x$.