Coordinates and Change of Basis

Subtopic: Vector Spaces

Different bases give different coordinate systems for the same vector space. The change of basis matrix translates coordinates from one system to another. This is essential for simplifying problems—choosing the right basis can turn a complicated matrix into a simple diagonal one.

Introduction

A map of a city uses a coordinate system. Switch from one map to another—say, from a tourist map to a surveyor's grid—and the same locations get different coordinate labels. The city hasn't changed; only the description has.

Similarly, a vector doesn't change when you switch bases, but its coordinates do. Understanding how coordinates transform lets you pick the basis that makes your problem easiest.

Coordinate Vectors

Let B={(v_1),(v_2),…,(v_n)} be a basis for V. Any vector x∈V can be uniquely written as:

x=(c_1)*(v_1)+(c_2)*(v_2)+⋯+(c_n)*(v_n)

The coordinate vector of x relative to B is:

[x]B=([(c_1)],[(c_2)],[⋮],[(c_n)])

This maps the abstract vector x to a concrete column of numbers.

Change of Basis Matrix

Suppose B={(v_1),…,(v_n)} and C={(w_1),…,(w_n)} are two bases for V. The change of basis matrix from B to C, denoted (P_C←B), satisfies:

[x]C=(P_C←B)*[x]B

To construct (P_C←B): express each basis vector of B in terms of C, and use those coordinates as columns.

(P_C←B)=([|,|,,|],[[(v_1)]C,[(v_2)]C,⋯,[(v_n)]C],[|,|,,|])

Worked Example

In (R^2), let
B={(1,0),(0,1)} (standard) and C={(1,1),(1,−1)}.

Find (P_C←B), the change of basis matrix from B to C.

Step 1: Express B vectors in terms of C

Write (1,0)=a(1,1)+b(1,−1).
This gives a+b=1 and a−b=0, so a=b=1/2.

Thus [(1,0)]C=(1/2,1/2).

Write (0,1)=a(1,1)+b(1,−1).
This gives a+b=0 and a−b=1, so a=1/2, b=−1/2.

Thus [(0,1)]C=(1/2,−1/2).

Step 2: Form the matrix

(P_C←B)=([1/2,1/2],[1/2,−1/2])

Step 3: Verify

Take x=(3,1). In standard basis B: [x]B=(3,1).

Apply the change of basis:

[x]C=(P_C←B)*[x]B=([1/2,1/2],[1/2,−1/2])*([3],[1])=([2],[1])

Check: 2*(1,1)+1*(1,−1)=(2,2)+(1,−1)=(3,1).

Key Properties

• (P_B←B)=I (identity — no change needed)

• (P_B←C)=((P_C←B))(−1) (reverse direction = inverse)

• (P_D←B)=(P_D←C)*(P_C←B) (chain rule for basis changes)

• Change of basis matrices are always invertible

Transformations in Different Bases

If T:V→V is a linear transformation with matrix A in basis B, its matrix in basis C is:

(A^′)=(P_C←B)*A*(P_B←C)=P(−1)*A*P

where P=(P_B←C). This is called a similarity transformation.

The power of choosing the right basis: if C consists of eigenvectors, then (A^′) is diagonal. This is diagonalization.

Applications

In computer graphics, changing between world coordinates, camera coordinates, and screen coordinates requires change of basis matrices.

In quantum mechanics, switching between position and momentum bases reveals different aspects of a quantum state.

In signal processing, changing from time domain to frequency domain (Fourier basis) simplifies analysis of periodic signals.

Summary

Coordinates depend on the choice of basis. The change of basis matrix (P_C←B) converts coordinates from basis B to basis C. Construct it by expressing B vectors in terms of C. The inverse goes the other direction. For linear transformations, changing basis transforms the matrix by (A^′)=P(-1)*A*P. The right basis choice can dramatically simplify a problem.