Linear dependence of linear functionals

Solution 1:

If $\alpha$ is the zero functional, we are done, because we take $k=0$.

Otherwise, consider a basis $\{v_\alpha\}$ of $V$. Let $\{v_p\}$ be the vectors that $\beta$ maps to nonzero scalars, and the $\{v_r\}$ the basis vectors mapped to zero. Then $\alpha$ must also map every vector in $\{v_r\}$ to 0, by hypothesis.

If $\{v_p\}$ contains just one vector we are done, because we can just scale $\beta$ so that $\alpha$ and $\beta$ agree on this basis vector. Otherwise, choose two vectors $v_1$ and $v_2$ in this set. Let $\beta(v_1)=b_1$, $\beta(v_2)=b_2$, $\alpha(v_1)=a_1$, and $\alpha(v_2)=a_2$. We want to show that $b_1/a_1=b_2/a_2$. Assume not. Then consider the vector $b_2v_1-b_1v_2$. We see that $\beta$ maps this to 0, but $\alpha$ does not, a contradiction.