UNI_FOUNDATION · Linear algebra · LA.BASIS.DIM

Basis, dimension and rank

Intuition

Understand first, compress later.

Linear algebra describes linear structure using vectors, matrices and transformations.

Switch representations

Basis and linear combinationv = (3, 2)

î = (1, 0)ĵ = (0, 1)v = 3·î + 2·ĵ

î and ĵ span the lattice: every lattice point is a whole-number combination of the two. The vector v is reached by laying 3 copies of î head to tail and then 2 copies of ĵ — which is exactly why the coordinates (3, 2) are the coefficients of that linear combination.

Formal view

You can basis, dimension and rank when you can recognise the underlying objects, select valid rules and check each step for equivalence or logical implication.

Why does it work?

This skill sits on a prerequisite network. If a step is unstable, repair the missing prerequisite instead of merely doing more items of the same type.