Linear Transformation Between Infinite Dimensional Vector Spaces

Question

Let there be a linear transformation $A$ that maps a vector in $\Bbb{R}^{\aleph _{0}}$ to a vector in $\Bbb{R}^{\aleph _{1}}$. In other words, $A$ maps a vector from a countably infinite vector space to an uncountably infinite vector space.

1. Is $AA^T$ invertible?
2. If $AA^T$ is symmetric then what is an upper bound for its smallest eigenvalue?