Dimension

Linear Algebra (Matrices) · Axiom Academy

Every basis for a space has the same number of vectors — and that number, the dimension, is what "size" really means. 1. Different Bases, Same Count Watch one basis of the plane morph into a completely different one. The arrows change — but they always sweep out the same , and there are always two of them. The count refuses to move. A totally different basis — still exactly two vectors Could some clever basis of use three vectors? Or one? The animation tests both. A third vector is always a combination of the others — dependent , not a basis. A single vector only sweeps out a line — it can't span . Exactly two survives. Any two bases of the same space have the same size More than vectors are forced to be linearly dependent — so no basis can be larger. Fewer than vectors cannot span V — so no basis can be smaller. Now read dimension as independent directions you can move . Climb the ladder inside : the origin alone, then a line, then a plane, then all of space. Each rung adds exactly one new direction — and the dimension ticks up by one. The standard basis of has n vectors, so Dimension is the one number every basis of a space agrees on — the count of independent directions, and the honest measure of how big the space is. Scroll up to revisit any step.

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