Inverse of an infinitely large matrix? This is probably a trivial problem for some people, but I've spent quite some time on it:
What is the inverse of the infinite matrix $$
\left[\begin{matrix}
0^0 & 0^1 & 0^2 & 0^3 & \ldots\\
1^0 & 1^1 & 1^2 & 1^3 & \ldots\\
2^0 & 2^1 & 2^2 & 2^3 & \ldots\\
3^0 & 3^1 & 3^2 & 3^3 & \ldots\\
\vdots & \vdots & \vdots & \vdots &\ddots
\end{matrix}\right]
$$
(Assume that $0^0=1$ for this problem).
I'm not sure if this problem has a solution or is well-defined, but if it has a solution, it would help greatly in a ton of stuff I'm doing (mostly related to generating functions and polynomial approximations). I began by taking the inverse of progressively larger square matricies, but I didn't see any clear pattern.

The linear transformations of $\Bbb R^\infty$ Take the form of row-finite matrices (all rows only have finitely many nonzero entries) or column finite matrices (analogously) depending on if you're writing the vectors on the right or left of transformations.
This matrix is neither row nor column finite, so it can't represent a linear transformation, invertible or not.
