 Randomized range finder

Choose a random sampling matrix $X$, form $Y=AX$, and find an orthonormal basis $Q$ for the sampled column space by a thin QR decomposition. The low-rank approximation $\hat{A}=Q{Q}^{T}A$ keeps the component of $A$ in that space. Since $Q{Q}^{T}AX=AX$, it reproduces the sampled action exactly. If $Y$ has deficient matrix rank, one can use only a basis of its column space; alternatively an orthonormal completion must be specified.