Change of Basis — Derived from the Definition and Visualized

Change of basis does not change the vector itself. It changes only the coordinates used to describe that vector. The first image derives the coordinate-change matrix from the definition of a basis. The following images show the same vector 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.

Explore more on GraphMath Linear Algebra

Change of basis derived from the definition

Starting with a square full-column-rank basis matrix A = [a₁ | a₂ | ... | aₙ], the change of coordinates from basis A to the standard basis is determined by how the basis vectors are mapped. Since A maps standard coordinate vectors to the basis vectors, the inverse A⁻¹ maps those basis vectors back to the standard coordinate directions. Click the image to open the full-size version.

Change of basis definition diagram deriving the coordinate transformation from a basis A to the standard basis, showing that A maps standard basis vectors to the basis vectors and A inverse maps basis vectors back to standard coordinate directions.

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 x_A means that the columns of A are the basis vectors, while x_A stores the coordinates relative to that basis. Click the image to open the full-size version.

Change of basis coordinate mapping diagram showing the same vector x in the standard basis and in the basis a1, a2, with x reconstructed as A x_A from its coordinates in the nonstandard basis.

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: x_B = B⁻¹A x_A and x_A = A⁻¹B x_B. Click the image to open the full-size version.

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 x_A is the coordinate column of a vector relative to that basis, then the actual geometric vector in standard coordinates is x = A x_A.

If B = [b₁ | b₂] is another basis matrix for the same space, then the same vector also satisfies x = B x_B. 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: Change of basis

Related work: GraphMath Linear Algebra

Contact: info@graphmath.com