Suppose that $P$ and $Q$ are two points in $\mathbb{P}^n(k)$. I have read in passing that there is a unique one-dimensional subspace of $\mathbb{P}^n(k)$ which contains both $P$ and $Q$ but I'm not convinced.
A one-dimensional subspace of $\mathbb{P}^n(k)$ corresponds to a $2$-dimensional subspace of $k^{n+1}$ which is of course a plane. A plane is not determined by two points however, it is determined by three points. Am I missing something or the result false?
On
Here is another way of looking at it. The projective space $\mathbb{P}^n$ is just normal Euclidean space with some points, lines, etc amounting to a copy of $\mathbb{P}^{n-1}$ added at infinity. So in geometric terms all this says is that between 2 point of $n$ dimensional space there is a unique line. You can hardly doubt that can you ?
A one dimensional subspace of projective space corresponds to a two dimensional subspace of affine space, i.e. a plane. A zero dimensional subspace, i.e. a point, of projective space similarly corresponds to a one dimensional subspace of affine space, i.e. a line.
You're right that a plane is not determined by two points, but it is determined by two lines.