Change of basis matrix with different dimensions

Question

Let's say I have 2 bases $A$ from $R^x$ and $B$ from $R^y$.

Is it possible to compute the change of basis matrix from $A$ to $B$ if $x \ne y$ ?

If $x = y$, it's easy, but if it is possible for $x \ne y$, how can I do it?

Answer 1

There does not exist an invertible linear transformation between two vector spaces of different dimension. Otherwise, this would be a homeomorphism, and two vector spaces (of finite dimension) are homeomorphic if and only if their dimension is the same. A change of basis matrix must be invertible by its definition (recall, you need to conjugate your matrix with the change of basis matrix, requiring that the change of basis matrix has an inverse: that's one way to see it), so it is impossible to construct a change of basis matrix between these vector spaces.

You can certainly create noninvertible maps between these two vector spaces which are invertible when restricted to a subspace of the larger vector space, allowing a "change of basis" between the subspace of the larger vector space and the smaller one by forcing the transformation to be between two vector spaces of the same dimension. You would do this in the same way as you would when $x = y$, but only defining your transformation on a restricted domain.