A transpose A invertible iff A invertible.

This video is somewhat involved and requires a basic understanding of matrix algebra and linear transformations. It proves a very nice result regarding matrices with linearly independent columns. Sal of Khan Acadamy provides a quality explanation.

The key takeaway is that when a matrix A has linearly independent columns, the product A^TA is guaranteed to be invertible. This connects directly to core ideas in Linear Algebra, especially the interplay between matrix transformations, null spaces, and invertibility. It shows how linear independence is not only preserved but reinforced through matrix operations, revealing deeper structure in how matrices behave.

This result is especially important in applications. The invertibility of A^TA underpins methods like least squares and the normal equations, which are fundamental tools in data fitting, optimization, and numerical analysis. In practice, this means that even if A itself isn’t square, we can still construct a system that behaves nicely and has a unique solution.

More broadly, this idea fits neatly within the Fundamental Theorem of Linear Algebra. The theorem links the column space, null space, row space, and left null space, showing how their structure determines properties like rank and invertibility. The fact that A^TA is full rank reflects these relationships, reinforcing how the geometry of these subspaces governs the behavior of matrix transformations.