A vague notion for sure, but is there a sense for linear transformations in which left adjoints give spanning vectors, right adjoints give independent vectors?

(the intuition is that those two can be "squeezed" into an invertible subspace)

Or if that's nonsense, here's a question:

**Puzzle AB1.** Prove that a linear transform \$$T \in \mathcal{L}(V,W)\$$ is a monotone map from linear subspaces in \$$(V, \subseteq_V)\$$ to linear subspaces in \$$(W, \subseteq_W)\$$.

**Puzzle AB2.** Given a linear transform \$$T \in \mathcal{L}(V,W)\$$ are there right and left adjoints of \$$T\$$ for linear subspaces in \$$(V, \subseteq_V)\$$ and \$$(W, \subseteq_W)\$$?

**Puzzle AB3.** How is this related to the pseudoinverse of a matrix?