When Systems Have No Solution

Linear Algebra (Matrices) · Axiom Academy

When can't be solved exactly, the best answer is the closest point we can reach — and geometry tells us exactly where it is. Sometimes there is no exact answer — only a best one A linear system has an exact solution only when lies inside the column space — the set of everything can reach. In the real world (fitting a line to scattered data, more equations than unknowns) almost never lands there. So we change the question: not "solve it exactly," but "get as close as we possibly can." Watch what "closest" means geometrically. is the tilted plane; the red point floats above it. Drop a perpendicular from straight down to the plane — where it lands is the nearest reachable point, , and the leftover gap meets the plane at a right angle. The closest point in the plane is the foot of the perpendicular from — that is the orthogonal projection of onto . The foot is with , and the residual has length . That residual is perpendicular to the whole plane — that perpendicularity is what makes the closest point. Why the foot, and nothing else, is closest Slide a point around inside the plane and watch the gap to . Every other reachable point makes a longer slanted gap; the distance shrinks to its minimum exactly when you reach the foot — the spot where the gap stands straight up. The smallest gap is the perpendicular one. Any sideways move trades a right angle for a slant — and a slant is always longer. Wherever the target goes, the answer is the projection

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