Change of Basis Matrices

Linear Algebra (Matrices) · Axiom Academy

LESSON The Change-of-Basis Matrix One matrix is the dictionary between two coordinate systems — its columns are the basis vectors, and that fact does all the work. 1. The Columns ARE the Basis Vectors Take the basis with , . Stack them as the columns of P_B . A coordinate vector is a recipe : take 3 of b_1 and 2 of b_2 . Multiplying by P_B lays those copies tip-to-tail and lands on x in standard coordinates. Columns of P_B are b_1, b_2 in standard coordinates P_B turns the recipe [x]_B into the actual vector x 2. The Inverse Reads Coordinates Back To go the other way — from a vector in standard coordinates to its B -coordinates — apply P_B^ -1 . Geometrically, this asks "how many of b_1 and b_2 rebuild x ?" by reading x off the slanted grid that b_1 and b_2 generate. Starting from , the answer is . Now bring in a second basis with , . The same point keeps its place; only its label changes. Relabel in two moves: P_B sends B -coordinates to standard, then P_C^ -1 reads standard into C -coordinates. Composed, that single matrix is . to standard, then into C — one matrix does both Worked example. Convert from basis B into basis C . Build the change-of-basis matrix from B to C as , where and . Apply it to the coordinate vector: . Check: , the same standard vector as in steps 1 and 2. You've seen the change-of-basis matrix as a coordinate dictionary: its columns are the basis vectors, and that single fact drives every conversion. Scroll up to revisit any step.

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