A system of linear equations admits two complementary geometric interpretations: the row picture, in which each equation defines a hyperplane (e.g., a line in 2D) and the solution is their point of intersection, and the column picture, in which the system is viewed as finding coefficients for a linear combination of the coefficient matrix's column vectors that produces the right-hand-side vector. These two pictures correspond to the same underlying matrix equation Ax = b, and the existence of a matrix inverse (A⁻¹A = I) provides a direct method for solving such systems. This is a foundational concept in linear algebra's treatment of linear systems, matrix equations, and vector space structure.
A system of linear equations admits two complementary geometric interpretations: the row picture, in which each equation defines a hyperplane (e.g., a line in 2D) and the solution is their point of i…