Change of Basis — Coordinate Mapping Visualized

Change of basis does not change the vector itself. It changes only the coordinates used to describe that vector. In the images below, the same vector is represented first in the standard basis and then in another basis. The coordinate column changes because the basis vectors change, but the geometric vector in the plane remains the same.

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Coordinates in one nonstandard basis

The same vector can be described either in the standard basis or in the basis (a₁, a₂). Writing x = A xA means that the columns of A are the basis vectors, while xA stores the coordinates relative to that basis.

A vector x is reconstructed as A x_A using the basis vectors a1 and a2. The diagram compares the standard basis e1, e2 with the basis a1, a2 and shows the same geometric vector represented in both coordinate systems.

Changing coordinates between two bases

The same vector can be tracked across two nonstandard bases by passing through standard coordinates. This makes the coordinate transformation explicit: xB = B−1A xA and xA = A−1B xB.

A comparison of three coordinate systems: basis b1, b2 on the left, the standard basis e1, e2 in the center and basis a1, a2 on the right. The same vector equals B x_B and A x_A, producing x_B = B inverse A x_A and x_A = A inverse B x_B.

Concept

Let A = [a₁ | a₂] be the matrix whose columns are the vectors of one basis. If xA is the coordinate column of a vector relative to that basis, then the actual geometric vector in standard coordinates is x = A xA.

If B = [b₁ | b₂] is another basis matrix for the same space, then the same vector also satisfies x = B xB. Since both expressions describe the same vector, their coordinates are related by a matrix transformation.

Structure

The key separation is between the vector itself and its coordinates. A basis matrix reconstructs the vector from coordinates. A change-of-basis matrix converts one coordinate column into another.

This representation separates basis reconstruction from coordinate transformation explicitly.

Reference

External reference: Wikipedia — Change of basis

Related chapter: Linear transformations

Related work: GraphMath Linear Algebra

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