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I would like some help with this exercise.

Suppose that $f_1,\ f_2 \in V^*$ and that $\text{Ker} f_1 = \text{Ker} f_2$. Show that $f_1 = k f_2$ for some scalar $k$.

I expect your suggestions.

Thanks!

JimmyJP
  • 394

1 Answers1

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If $\ker f_1=\ker f_2=V$ then the result is trivial. Now assume that $\ker f_1\ne V$ and let $a\not\in \ker f_1$ such that $f_2(a)=1$. Let $k=f_1(a)$. We have easily $$V=\ker f_1\oplus\operatorname{span}(a)$$ so let $x=x_1+\alpha a$ where $x_1\in\ker f_1$ then

$$f_1(x)=f_1(\alpha a)=\alpha f_1(a)=\alpha k=k\alpha f_2(a)=kf_2(\alpha a)=kf_2(x)$$