Topic: Fundamental Subspaces and Rank

Why Row Rank Equals Column Rank

Why Pivot Columns Must Come from the Original Matrix

What Row Operations Preserve—and What They Do Not

Using Rank-Nullity to Recover Unknown Dimensions

Upper Bounds for the Rank of a Product

The Matrix Rank-Nullity Theorem

The Extended Invertible Matrix Theorem

Testing Equivalent Conditions in the Invertible Matrix Theorem

Sylvester’s Rank Inequality

Row Space and Equations on Coordinates