Coordinates and Change of Basis: Explained with Examples

Coordinates are a vector's address relative to a chosen basis: if v = c1 b1 + … + cn bn, its coordinate vector is the column (c1, …, cn). The same arrow gets completely different coordinates under a different basis, just as one location looks different in different map projections.

Changing basis matters because the right basis simplifies hard problems. Diagonalising a matrix, compressing an image or solving a differential equation all amount to choosing coordinates where the action looks simple. The change-of-basis matrix is the dictionary translating between descriptions without losing information.

Key Concepts

• Coordinate vector [v]B: scalars expressing v in the ordered basis B.
• Change-of-basis matrix PB to C: columns are the B basis vectors written in C coordinates; it converts via [v]B = PB to C [v]C.
• Direction warning: the matrix is built from B vectors in C coordinates but maps coordinate vectors the opposite way, from C to B.

Formulas

• Conversion: [v]B = PB to C [v]C, column j being [bj]C.
• Inverse relationship: PC to B = (PB to C) inverse, and their product is the identity.
• Standard shortcut: if C is standard, the columns of PB to C are just the B vectors themselves.
• Consistency check: converting there and back must return the original coordinates.

Worked Example

In R squared let C = (c1, c2) with c1 = (1,1), c2 = (1,minus 1), and E the standard basis. Then PE to C has columns (1,1) and (1,minus 1), i.e. [[1,1],[1,minus 1]]. Take [v]C = (3,2), meaning v = 3c1 + 2c2. Convert: [v]E = PE to C (3,2) = (3+2, 3 minus 2) = (5,1). Direct check: 3(1,1)+2(1,minus 1) = (5,1), matching. Going back, det = minus 2, so the inverse is [[1/2,1/2],[1/2,minus 1/2]]; applied to (5,1) it gives ((5+1)/2, (5 minus 1)/2) = (3,2), recovering the original coordinates exactly.

Common Misconceptions

• The matrix maps old coordinates to new. Reversed: PB to C takes C-coordinates to B-coordinates. Columns are new-basis vectors in old coordinates; conversion runs the other way.
• Coordinates belong to the vector alone. No coordinates exist until a basis is chosen; (5,1) in E is (3,2) in C, same arrow.

Real-World Applications

• Graphics: world, camera and screen coordinates connect through chains of change-of-basis matrices.
• Physics: aligning axes with a problem's symmetry, like forces on an incline, is a change of basis.
• Data science: PCA changes to a basis of uncorrelated, variance-ordered coordinates.

Coordinates are always relative to a basis, and the change-of-basis matrix translates between choices, its columns holding the new basis in the old language. Build it column by column, mind the direction, and verify with the inverse.