Exercise 17

Example 2.3 shows that changing the pair of bases can change the map that a matrix represents, even though the domain and codomain remain the same. Could the map ever not change? Is there a matrix $H$, vector spaces $V$ and $W$, and associated pairs of bases $B_{1},D_{1}$ and $B_{2},D_{2}$ (with $B_{1}\neq B_{2}$ or $D_{1}\neq D_{2}$ or both) such that the map represented by $H$ with respect to $B_{1},D_{1}$ equals the map represented by $H$ with respect to $B_{2},D_{2}$?