Computing Change of Basis

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Computing a Change of Basis Build the change-of-basis matrix and use it to convert a vector's coordinates from one basis to another. Let and be two bases of . A vector has coordinates relative to . Find its coordinates relative to . Nice work! You built a change-of-basis matrix from scratch and used it to re-express a vector in a new basis. Here is what carried the calculation: Bases become matrices: writing each basis's vectors as columns gives and , the matrices that turn coordinates relative to that basis into standard coordinates. The change-of-basis matrix: converts -coordinates straight into -coordinates — go to standard with , then come back out with . Reverse with the inverse: to go the other way, . A change-of-basis matrix is always invertible. Check by reconstructing : and — same vector, two coordinate descriptions. The same vector looks different in different bases, but it is one geometric object. Change of basis is the dictionary between those descriptions — central to graphics, data science, and quantum mechanics.

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