I have this problem I'm working on, which I cannot entirely solve:

Let $V$ be a finite dimensional vectorspace over a field $K$ with a symmetric bilinear form $\langle \cdot, \cdot \rangle$. Define for every $v \in V$ the map $$ l_v : V \mapsto K: w \mapsto \langle v, w \rangle. $$ Consider now the map $$f: V \rightarrow V^{*}: v \mapsto l_v. $$

(i) Prove that $f$ is a linear map.

(ii) Prove that $f$ is surjective if and only if $\langle \cdot, \cdot \rangle $ is non degenerate.

Attempt: (i) Let $v, w \in V$ be vectors, and let $\lambda, \mu \in K$ be scalars. We need to prove that $$f(\lambda v + \mu w) = \lambda f(v) + \mu f(w). $$ This is equivalent to proving $$l_{\lambda v + \mu w} = \lambda l_v + \mu l_w. $$ Let $x \in V$ be another vector. Then the above equality is true since $$ l_{\lambda v + \mu w} (x) = \langle \lambda v + \mu w, x \rangle = \lambda \langle v, x \rangle + \mu \langle w, x \rangle = \lambda l_v(x) + \mu l_w(x). $$ This proves that $f$ is linear.

(ii) Suppose first that $f$ is surjective. To prove that $\langle \cdot, \cdot \rangle$ is non degenerate, we need to prove that $$ \forall w \in V: [ (\forall v \in V: \langle v, w \rangle = 0 ) \Rightarrow w = 0 ]. $$ So let $w \in V$ be arbitrary, and suppose that $\forall v \in V: \langle v, w \rangle = \langle w, v \rangle = 0$, since the bilinear form is symmetric. This means that $\forall v \in V : l_v(w) = l_w(v)= 0. $

Now I'm not sure how to use the fact that $f$ is surjective to deduce from this that $w = 0$. I know that the linear functional $l_v \in V^{*}$. So since $f$ is surjective, I can take a $v \in V$ such that $f(v) = l_v$. But what can I conclude from this about $w$?

Help is appreciated.

  • 2
    $\begingroup$ Is $V$ finite dimensional ? $\endgroup$ – Dark Aug 15 '16 at 15:46
  • 1
    $\begingroup$ Yes, it is finite dimensional. $\endgroup$ – Kamil Aug 15 '16 at 16:16

If $V$ is finite dimensional (if!), then you can use the rank-nullity theorem which tells you that $f$ is injective since it is surjectitve. Thus, continuing at one of your final steps, $$ l_w(v) = 0\;\forall v\in V\;\Rightarrow\;f(w) = 0 \;\Rightarrow \; w=0 $$ as desired.


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