Exercise from PR Halmos's Finite-Dimensional Vector Spaces, Edition 2 (Ex 8 on page 142).
If $x$ and $y$ are nonzero vectors (in a finite-dimensional inner product space) then a necessary and sufficient condition that there exist a positive (nonnegative semidefinite) transformation $A$ such that $Ax = y$ is that $(x, y) > 0$.
As part of establishing the necessity, I have managed to establish the inequality $(x, y) \geq 0$ using the positive character of $A$ which implies $(Ax, x) \geq 0$. However, I have not progressed to establish the strict inequality. In particular, I am not sure how I should exploit the finite-dimensional character of the inner product space.
As for establishing the sufficiency, I am (far) more uncertain. In particular, if I just write $y = Ax$ (as a general map, not necessarily a linear transformation), then $(Ax, x) >0$ (strictly positive) follows from the hypothesis $(x, y) > 0$. However, it is unclear how to establish that $A$ is a linear transformation and that it is self-adjoint as well (i.e, $(Az, w) = (z, Aw)$ holds for all vectors).
Would appreciate a hint or solution.
Hint: $A=cP$ where $c$ is a suitable positive constant and $P$ is the projection on the one-dimensional space spanned by $y$ does the trick.