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M02.9 · Change of basis

A free, illustrated module of 4 lessons and one interactive in M02 Linear Algebra I: Vectors and Matrices: Coordinates depend on the basis; The change-of-basis matrix; The same map in a new basis; Choosing a good basis.

M02.9 L1 · Core 4 lessons · about 100 min Live · in review

Part of M02 Linear Algebra I: Vectors and Matrices in the Mathematics track.

The numbers that name a vector are a recipe written for one grid, and a different grid gives the same arrow different numbers. These four lessons read arrows on a second grid, pack the conversion into one matrix and its inverse, rewrite a whole linear map for a new basis with the three-step rule “translate in, act, translate out”, and end by choosing the basis in which a map becomes a plain stretch: the idea behind eigenvectors and principal component analysis.

What you need first. Basis and unique coordinates (M02.2), the matrix-vector product as a combination of columns (M02.4), matrix multiplication as composition and powers (M02.5), solving a small linear system (M02.6), the inverse of a matrix (M02.7) and the determinant with its product rule (M02.8).

Lessons

Coordinates depend on the basis M02.9.1 25 min

The numbers that name a vector are a recipe written for one particular grid. Learn to read the same arrow on a second grid, find its new coordinates by solving two small equations, and see why no single list of numbers is the vector itself.
The change-of-basis matrix M02.9.2 25 min

Put the new basis vectors in the columns of a matrix P. Multiplying by P turns new coordinates into standard ones, and multiplying by its inverse turns them back. Learn why, work both directions with numbers, and see why the direction feels backwards at first.
The same map in a new basis M02.9.3 25 min

The matrix of a linear map depends on the basis used to write it. Derive the rule that the new matrix is P inverse times A times P by translating in, acting and translating out, work it by hand for the course’s running map, and prove that the determinant does not change.
Choosing a good basis M02.9.4 25 min

Some bases make a map look simple. Find the basis in which a map of the plane is nothing but a stretch along two directions, watch its matrix turn diagonal, and see how the same move gives fast powers, eigenvectors and principal component analysis.

The interactive

M02.9-X1 · Two-grid viewer. Drag an arrow and the two vectors of a second basis. Both sets of coordinates, the change-of-basis matrix and its inverse update as you move, and the arrow is rebuilt from its new coordinates as a check. It lives in The change-of-basis matrix.