What's the most intuitive Geometric proof that makes it almost obvious or trivial.. rank(A) = rank(A^T)?
I already understand the algebraic proofs using the Fundamental Theorem of Linear Algebra, Rank-Nullity Theorem, Gaussian elimination, etc. Those are clear to me.
What I'm looking for is the deep intuition behind why this has to be true.
In other words, why is the dimension of the column space always equal to the dimension of the row space of the same matrix?
Geometrically, the column space and row space live in different vector spaces (R^m vs. R^n), so it isn't obvious to me why they should always have the same number of independent directions. What is the underlying constraint that forces this equality?
I'm not looking for another algebraic derivation. Instead, I'd love explanations that answer questions like:
What is the geometric picture?
Is there an information-theoretic, transformational, or degrees-of-freedom interpretation that makes this equality feel inevitable rather than something we simply prove algebraically?
Are there any visualizations or mental models that make this theorem "click"?
I'm especially interested in explanations that make the result feel almost obvious once you see the right perspective.
Edit:
I know most of the popular formal algebraic proofs to prove this, what i am looking for is intuitive perspective
For example, we can intuitively understand why
rank(A) + nullity(A) = n
When we apply the transformation A to vectors, each independent direction has only two possibilities: it either survives (maps to a nonzero independent direction) or it is killed (maps to the zero vector). Since these are the only two outcomes for the n independent input directions, it is intuitive that
rank(A) + nullity(A) = n
I'm looking for a similarly intuitive explanation for this theorem. Rather than an algebraic proof, I want a geometric or conceptual way to understand why it must be true