Building with Vectors

Linear Algebra (Matrices) · Axiom Academy

Scale a few vectors, add them up, and watch which points you can reach. That reachable set has a name — the span . One move builds an entire space Give linear algebra two arrows and one rule — scale each one and add them — and out comes a whole space. That single move, the linear combination , is the engine under solving systems, coordinates, and dimension. Before any of the machinery, just watch it run. Watch build tip-to-tail: a scaled copy of , then a scaled copy of laid on its tip, landing the purple result. As the scalars c_1,c_2 sweep through every value, that landing tip rakes across the plane — and the set of all the places it can land is the span . Every purple dot is one linear combination. Sweep the scalars and they fill the plane — that filled region is . Drive the scalars, reach a target Now you hold the dials. Slide c_1 and c_2 : a scaled and a scaled build the result tip-to-tail, leaving a trail of every point you visit. Steer the purple tip onto the gold target (4,5) — when you land it, you've written (4,5) as a linear combination, so it lives in the span. Land it and the readout turns green: . Reachable means "in the span." How much can two vectors reach? Does reach the whole plane, or only a line through the origin? It hinges on one thing: independence . Swing around. While it points a genuinely different way, the span is the full plane. Line it up with and the second vector adds nothing new — every combination collapses onto a single line.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.