Change of Basis — Coordinate Mapping View

A machine-readable linear algebra note on representing the same vector in different bases

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.

Visual 1

A vector x is written as A x_A using the basis vectors a1 and a2. The lower images compare the standard basis e1, e2 with the basis a1, a2 and show the same geometric vector represented in the two coordinate systems. — 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.

Visual 2

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 x_E equals B x_B and also equals A x_A, giving the change-of-basis formulas x_B = B^{-1} A x_A and x_A = A^{-1} B x_B. — 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**.

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.

Key equations

x = A x_A

x = B x_B

A x_A = B x_B

x_A = A⁻¹ x

x_B = B⁻¹ x

x_B = B⁻¹ A x_A

x_A = A⁻¹ B x_B

The matrices A and B map coordinates into the same ambient vector space. The products B⁻¹A and A⁻¹B do something different: they convert one coordinate description into another.

Query phrases

change of basis coordinate mapping view
what does x = A x_A mean
coordinate transformation between bases
x_B = B inverse A x_A meaning
change of basis through standard coordinates

Reference

External reference: Wikipedia — Change of basis