Minimal Spanning Sets

Linear Algebra (Matrices) · Axiom Academy

A set of vectors can span all of with vectors to spare. Strip away the ones you don't need, and the leanest set that still spans the whole space is exactly a basis. In you only ever need two good vectors to reach every point in the plane. Hand someone a spanning set with extra vectors and some of them are just dead weight — you can throw them out and lose nothing. The set you're left with, once nothing more can go, is a basis . Watch four vectors that together span the whole plane (shaded). Two of them are redundant — each is already a combination of the others — so as they're removed one at a time, the shaded span never budges. What settles is the minimal pair : remove either of those and the plane would collapse. The redundant vectors fall away and the span holds steady — proof they were never carrying any of it. Two vectors do all the spanning. Which vectors are safe to drop? Here are the same four vectors. Click any vector to toggle it out of the set and watch the shaded span respond. Some removals are safe — the vector was redundant, so the plane stays whole. Others break the span , dropping the plane down to a line. Find every vector you can throw away. Click a vector to drop it in or out: A vector you can remove without shrinking the span is redundant — it's already a combination of the others. A vector whose removal collapses the span is essential .

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