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(x0) is not zero for some vector x0 in V. Then the only possibility for a is a = y(x0)/z(x0). Now I need to prove the identity for all x in V:
y(x) = y(x0)z(x)/z(x0)
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.