We again need to prove both directions of an “if and only if”. If an isomorphism exists, can you see how to use
Exercise 2.2.13 to show the dimensions are equal?
If the dimensions are equal, you need to construct an isomorphism. Since
and
are finite-dimensional, you can choose a basis for each space. What can you say about the sizes of these bases? How can you use them to define a linear transformation? (You might want to remind yourself what
Theorem 2.1.8 says.)