The core theoretical framework established is the structural decomposition of rectangular matrices into four mutually orthogonal subspaces: row space, column space, null space, and left null space. Formally defined by rank-nullity theorem constraints ($r + \dim(\text{Nul } A) = n$), these spaces partition $\mathbb{R}^n$ such that the intersection of any two distinct spaces (e.g., Row Space and Null Space) contains only the zero vector, establishing a geometric orthogonality fundamental to linear system solvability. This concept anchors functional analysis within finite-dimensional geometry by defining solution sets as affine sums ($\mathbf{x}_p + \text{Nul } A$) of particular solutions and homogeneous null spaces determined by pivot column independence.
The core theoretical framework established is the structural decomposition of rectangular matrices into four mutually orthogonal subspaces: row space, column space, null space, and left null space. F…