A vector is geometric; its coordinate list depends on the basis used to describe it. This chapter shows how a basis matrix converts basis coordinates into standard coordinates, how its inverse converts standard coordinates back into basis coordinates and how the same idea extends to dimension-lowering coordinates and projections.
Change of basis separates a geometric vector from the coordinates used to describe it.
The chapter uses both 2D coordinate grids and the dimension-lowering case to show that basis choice changes coordinates, not the underlying vector.
The basis matrix A maps A-coordinates back to the original vector: x = AxA. Therefore the inverse map sends the original vector to its A-coordinates: xA = A⁻¹x.
The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.
| Topic | Pages |
|---|---|
| Change of basis: definition | 1–2 |
| Same vector in two coordinate systems | 3–5 |
| Change of basis with dimension lowering | 5–6 |
| Why change basis? | 6–7 |
A tall full-column-rank matrix U does not have a two-sided inverse. Instead, U⁺ = (UᵀU)⁻¹Uᵀ gives the left inverse that extracts coordinates in the u-basis, and the round trip UU⁺ returns the projection onto col(U).