Simple question about linear functionals

Hi everyone. Paul Halmos asks a seemingly-innocuous question in Finite Dimensional Vector Spaces: let V be a vector space over a field F and let y and z be linear functionals from V to F. Suppose, furthermore that y(x) = 0 whenever z(x) = 0, for all x in V. Prove that y = az for some scalar a.

Here's where I am so far: if z = 0 then the problem is trivial, so without loss of generality z(x_{0}) is not zero for some vector x_{0} in V. Then the only possibility for a is a = y(x_{0})/z(x_{0}). Now I need to prove the identity for all x in V:

y(x) = y(x_{0})z(x)/z(x_{0})

But I don't see how to do this. I have absolutely no background with linear functionals, and only know undergraduate-level linear algebra. I'd appreciate a tip.

Re: Simple question about linear functionals

Write $\displaystyle x=x-\frac{z(x)}{z(x_0)}x_0+\frac{z(x)}{z(x_0)}x_0$ and compute $\displaystyle y\left(x-\frac{z(x)}{z(x_0)}x_0\right)$.

Re: Simple question about linear functionals

I'm having trouble even with the hint. I would love to compute $\displaystyle y\left(x-\frac{z(x)}{z(x_0)}x_0\right)$. Indeed, this quantity needs to be zero, and then the result immediately follows. But showing that this result is equal to zero is really a trivial modification of the problem:

$\displaystyle y\left(x-\frac{z(x)}{z(x_0)}x_0\right)=y(x)-y\left(\frac{z(x)}{z(x_0)}x_0\right)$

$\displaystyle =y(x)-\frac{y(x_0)}{z(x_0)}z(x)$

$\displaystyle =(*)$

is precisely the quantity I need to show is zero. I don't see any way to "compute" this further, so that's kind of my problem. I could give it a try:

$\displaystyle (*)=\frac{1}{z(x_0)}\left(y(x)z(x_0)-y(x_0)z(x)\right)$

$\displaystyle =\frac{1}{z(x_0)}z\left(y(x)x_0-y(x_0)x\right)$

Basically I'm just trying to use linearity here. I am not aware of any other possible manipulations (short of adding and subtracting a quantity which I just don't see). Could I please have a bit more help?

Re: Simple question about linear functionals

Ah hah. I see what I needed to do. I do not need to evaluate y(...). I need to show that z(...) = 0, and then by hypotheses, y(...) = 0. But showing that z(...) = 0 is very easy. That solves the problem. Thank you. I was being silly - I should have looked for a way to use the given hypotheses.

Thanks again! And by the way, a great hint that didn't give away too much of the fun!