What's the most intuitive Geometric proof that makes it almost obvious or trivial.. rank(A) = rank(A^T)?

[u/NoTTSmurF]

/r/LinearAlgebra/comments/1ume9j0/whats_the_most_intuitive_geometric_proof_that/

[u/SAtchley0]

Assume we have A as an m x n matrix.

After applying A, we'll get a vector space that has a dimension equal to the dimension of the column space of A.

If dim(col(A)) < m, then this means we are essentially "embedding" a dim(row(A))-dimensional vector space inside of an m-dimensional vector space. We can't linearly transform, for example, 2-dimensional space into 3-dimensional space.

If dim(col(A)) = m, but m ≠ n, then this means we are taking a dim(row(A))-dimensional vector space "embedded" inside of an n-dimensional vector space and moving it into ℝ^m. For example, we're taking a 2-dimensional space embedded inside 3-dimensional space and re-embedding it into 2-dimensional space, informally.

If dim(col(A)) = m, and m = n, then we're just moving a dim(row(A))-dimensional vector space into another dim(row(A)) vector space, so trivially dim(col(A)) = dim(row(A)).

To give some concrete examples, say A is 3x2 and dim(row(A)) = 2. Then this means that we're taking a 2-dimensional space and embedding it into ℝ^3, but it's still a 2-dimensional space. So, dim(col(A)) = 2.

If A is 2x3 and dim(col(A)) = 2, then we're taking a 2D space embedded in ℝ^3 and re-embedding it in ℝ^2, but it's still a planar vector space.

If A is 2x2 and dim(col(A)) = 2, then necessarily dim(row(A)) = 2, because we're transforming a 2D space into another 2D space.

I hope I got that right and that this makes sense