I'm currently going through Artin's Abstract Algebra book, and I'm a bit confused about the following:
He says that a form $<,>$ on a finite dimensional vector space $V$ is non-degenerate if its nullspace is $\{0\}$.
Now, here is my question:
There are two potential meanings for the notation $W^\perp$:
$W^\perp$ may denote the orthogonal space to $W$. That is: $$W^\perp = \{v \in V: <v,w> = 0 \space \text{for all}\space w \in W \}$$
$W^\perp$ may denote the nullspace of the restriction $<,>|_W$. That is: $$W^\perp = \{v \in W: <v,w> = 0 \space \text{for all} \space w \in W \}$$
When we say that a form $<,>$ on $V$ is non-degenerate on a subspace $W$, we are trying to say that $W^\perp = \{0\}$, but which interpretation of $W^\perp$ is normally meant?
The second one:$$W^\perp=\left\{v\in W\,\middle|\,(\forall w\in W):\langle v,w\rangle=0\right\}.$$