Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 1 / 8B / a / iii / Solution

Rescaling the first basis vector gives the change-of-basis matrix $P=\mathrm{diag}(k,1)$, invertible because $k\ne 0$. Direct multiplication givesThus the matrices are related by matrix similarity. In terms of basis images, the coefficient along the rescaled first basis vector is divided by $k$, while the second component of its image is multiplied by $k$.