Basis Definition

Linear Algebra (Matrices) · Axiom Academy

A basis is a set of vectors that is both linearly independent and spans the space — just enough to reach everything, with no redundancy. 1. A Basis Must Pass Two Tests To be a basis for a space, a set of vectors has to clear two hurdles at the same time. Watch three candidate sets for the plane get put through both tests — only one passes. Independent and spans — its combinations fill the whole plane. Passes both. ✓ Independent, but one vector only reaches a single line — it doesn't span. Fails test 2. Spans the plane, but adds nothing new — the set is dependent . Fails test 1. Drop a vector and you lose reach; add a redundant one and you lose independence. A basis sits exactly in between. 2. Just Enough: Minimal Spanning, Maximal Independent There's a tidier way to say "independent and spanning." A basis is the smallest set that still spans, and equivalently the largest set that stays independent. Watch the set grow one vector at a time and see where it's "just right." stays independent but only spans a line. Add to reach more. Not yet spanning. spans all of and is still independent. This is a basis : minimal spanning and maximal independent. A third vector in the plane must already be a combination of the first two — it breaks independence without adding reach. "Smallest spanning set" and "largest independent set" describe the same set. That set is the basis. 3. Every Vector Has Exactly One Address

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