There is a question that I don't understand it well enough.
Let $A \in \mathbb{C}^{n\times n}$. Show that $W(A) = \{\langle Ax,x\rangle: x\in \mathbb{C}^n, ||x|| \leq 1\}$ if and only if $0 \in W(A)$ where $W(A)$ is the numerical range.
The definition in the literature: $W(A) = \{\langle Ax,x\rangle: x\in \mathbb{C}^n, ||x|| =1\}$.
Thanks for any idea/explanation.
Let $B=\{\langle Ax,x\rangle: x\in\mathbb C^n, \|x\|\le1\}$. The question asks you to prove that $B=W(A)$ if and only if $0\in W(A)$.
The "only if" part is trivial: if $B=W(A)$, clearly $0=\langle A0,0\rangle\in B=W(A)$.
For the "if" part, suppose $0\in W(A)$. Since $W(A)$ is convex, $$ \langle A(cx),(cx)\rangle=(1-c^2)0+c^2\langle Ax,x\rangle\in W(A) $$ for any unit vector $x$ and any $0\le c\le1$. It follows that $B\subseteq W(A)$. But by definition, $B$ is a superset of $W(A)$. Hence $B=W(A)$.