Let \(T\) be a linear transformation from \(\RR^3\) into \(\RR^2\), and let \(U\) be a linear transformation from \(\RR^2\) into \(\RR^3\). Prove that the transformation \(UT\) is not invertible. Generalize the theorem.
Solution
The composite \[ UT:\mathbb R^3\longrightarrow\mathbb R^3 \] factors through \(\mathbb R^2\). Hence \[ \operatorname{rank}(UT)\le \operatorname{rank}(T)\le 2<3. \] An invertible operator on \(\mathbb R^3\) must have rank \(3\), so \(UT\) is not invertible.
More generally, let \(V,W\) be finite-dimensional vector spaces with \(\dim W<\dim V\), and let \[ T:V\to W,\qquad U:W\to V \] be linear. Then \[ \operatorname{rank}(UT)\le \dim W<\dim V, \] so \(UT\) cannot be invertible on \(V\).