The following is a good illustration:
Let $P$ be a finite set ("points"), and let $L\subset 2^P$ ("lines") be such that distinct lines intersect in at most one point and any two distinct points are contained in a line. Let $V$ be the real vector space with basis $P$, $W$ the vector space with basis $L$. There are natural linear maps $T\colon V\to W$ and $S\colon W\to V$ mapping every point to the sum of the lines containing it, and every line to the sum of the points in it. Then $ST = J+D-I$ where $J$ is the all-ones matrix (through every two distinct points there is a unique line), $I$ the identity matrix and $D$ is diagonal with entries counting the lines through each point.
Assume that not all points are collinear. Then all the diagonal entries of $D-I$ are at least one; it is then easy to verify that the determinant of $ST$ is positive, and conclude that $|L| \geq |P|$.
