Mathematics.

vector spaces

Change of Basis

Linear Algebra30 minDifficulty5 out of 10

You should know: vector space basis and dimension, linear transformation

Overview

The same vector can be described by different coordinate lists depending on which basis is used, and a change-of-basis matrix converts coordinates from one basis to another without changing the underlying vector itself. If B is a basis whose vectors are the columns of a matrix P, then a vector's B-coordinates c relate to its standard coordinates x by x = Pc, so converting the other way requires c = P⁻¹x. Change of basis is essential for simplifying linear transformations — expressing a transformation matrix in the 'right' basis (such as an eigenbasis) can turn a complicated matrix into a simple diagonal one.

Intuition

Coordinates are just labels that depend on which 'ruler and axes' you chose. If you switch from the standard x-y axes to a new set of (possibly skewed) basis axes, the same physical point gets different coordinate numbers. The change-of-basis matrix P is a translator: multiplying new-basis coordinates by P gives standard coordinates, and multiplying by P⁻¹ goes the other way. The real payoff is for transformations: a matrix that looks complicated in standard coordinates can become diagonal — pure independent scaling — when re-expressed in the basis of its own eigenvectors, exactly the idea behind diagonalization.

Formal Definition

Definition

If B = \{b_1,\ldots,b_n\} is a basis of V and P = [b_1\ \cdots\ b_n] is the matrix of basis vectors as columns, then for a vector v with standard coordinates x and B-coordinates [v]_B:

x=P [v]B⟺[v]B=P−1xx = P\,[v]_B \qquad \Longleftrightarrow \qquad [v]_B = P^{-1} x
Change of basis for coordinate vectors
[T]B=P−1AP[T]_B = P^{-1} A P
Matrix of a transformation A rewritten in basis B

Worked Examples

  1. Form P from the basis vectors as columns, and multiply by the B-coordinate vector.

    P=(111−1),x=P(23)P = \begin{pmatrix}1&1\\1&-1\end{pmatrix}, \quad x = P\begin{pmatrix}2\\3\end{pmatrix}

  2. Compute the matrix-vector product.

    x=(1(2)+1(3)1(2)+(−1)(3))=(5−1)x = \begin{pmatrix}1(2)+1(3)\\1(2)+(-1)(3)\end{pmatrix} = \begin{pmatrix}5\\-1\end{pmatrix}

Answer: Standard coordinates: (5, -1).

Practice Problems

Difficulty 5/10

Given basis B = {(2,0), (0,1)} of ℝ², find the standard coordinates of the vector with B-coordinates (3, 4).

Difficulty 6/10

Using B = {(2,0),(0,1)}, find the B-coordinates of the standard vector (6,4).

Difficulty 7/10

A matrix A represents a transformation that's complicated in standard coordinates but becomes diag(3,2) when expressed in the eigenbasis P = [[1,1],[0,-1]] (as in the diagonalization worked example). Explain the relationship [T]_B = P⁻¹AP in this context and why engineers prefer working in the eigenbasis for repeated applications of A.

Quiz

If P is the matrix whose columns are a basis B's vectors, then for a vector v with B-coordinates [v]_B, the standard coordinates x satisfy:
Rewriting a linear transformation matrix A in a new basis B (with basis matrix P) gives:
Choosing the eigenbasis as the new basis for a diagonalizable matrix A results in [T]_B being:

Summary

  • A change-of-basis matrix P (columns = new basis vectors) converts coordinates: x = P[v]_B and [v]_B = P⁻¹x.
  • A linear transformation's matrix changes under a change of basis via the similarity transform [T]_B = P⁻¹AP.
  • Choosing the eigenbasis as the new basis diagonalizes A, turning repeated transformation into simple independent scaling.

References

  1. WebsiteWikipedia — Change of basis