A basic fact of linear algebra, which is surprisingly useful is \[V\subset W \iff V^{\perp}\supset W^{\perp}.\]

This is true because the linear functionals annhiliating \(V\) are less specific than the linear functionals annhilating both \(V\) and \(W\setminus V\).

But this can be useful, for example, to show that a vector is in some space.

Do you have any cute facts that are obvious but also super useful? Let me know!