Change of Basis Matrix Calculator

Calculate transformation matrices between different bases of vector spaces. Supports complex numbers, up to 10 dimensions, and customizable precision.

Calculator Settings

Real Complex
4

Change of Basis Formula: Given two bases B and B' of a vector space V, the change of basis matrix P from B to B' satisfies: [v]_{B'} = P [v]_B

Where: [v]_B are coordinates of vector v in basis B, and [v]_{B'} are coordinates in basis B'.

Each basis must contain exactly n linearly independent vectors (1-10 dimensions supported).
Basis B (Original)
Basis B' (New)
Vector Coordinates
Enter each basis vector as a column. For ℝ³, vectors look like: [1, 0, 0], [0, 1, 0], [0, 0, 1]
Standard Basis (ℝⁿ)
Random Basis
Example 1
Example 2
Complex Example
Calculating...

Understanding Change of Basis

In linear algebra, a change of basis is the process of expressing vectors in a vector space with respect to a different basis. The change of basis matrix allows us to convert coordinates from one basis to another.

Mathematical Definition:

Let B = {b₁, b₂, ..., bₙ} and B' = {b'₁, b'₂, ..., b'ₙ} be two bases for the same vector space V.

The change of basis matrix P from B to B' is defined by:

[v]_{B'} = P [v]_B

where [v]_B and [v]_{B'} are the coordinate vectors of v with respect to bases B and B' respectively.

How to Compute the Change of Basis Matrix

1

Form basis matrices: Create matrices B and B' whose columns are the basis vectors of B and B' respectively.

2

Compute the transformation matrix: The change of basis matrix from B to B' is P = (B')⁻¹B.

3

Inverse transformation: The change of basis matrix from B' to B is the inverse: Q = B⁻¹B' = P⁻¹.

Properties of Change of Basis Matrices

  • Invertibility: Change of basis matrices are always invertible (since bases consist of linearly independent vectors).
  • Composition: Changing from basis B to B' and then from B' to B'' corresponds to multiplying the change of basis matrices.
  • Linear Transformation Matrix: If T: V → V is a linear transformation with matrix A relative to basis B, then its matrix relative to basis B' is P⁻¹AP.
  • Determinant: The determinant of a change of basis matrix is nonzero.

Examples of Common Basis Changes

Standard basis to another basis:
In ℝ², from standard basis {e₁, e₂} to basis {(1,2), (3,4)}:
1
3
2
4
Rotation of axes in ℝ²:
Rotating by angle θ: basis { (cos θ, sin θ), (-sin θ, cos θ) }
cos θ
-sin θ
sin θ
cos θ

Applications of Change of Basis

  • Diagonalization: Finding a basis in which a linear transformation is represented by a diagonal matrix
  • Computer Graphics: Transforming between different coordinate systems (world, camera, screen coordinates)
  • Quantum Mechanics: Changing between different representations of quantum states
  • Signal Processing: Transforming between time domain and frequency domain representations
  • Principal Component Analysis (PCA): Finding the most informative basis for data representation

Calculator Features:

  • Handles vector spaces of dimensions 1-6
  • Computes both directions of basis transformation (B to B' and B' to B)
  • Verifies calculations by showing that P × P⁻¹ = I
  • Optionally converts vector coordinates between bases
  • Uses math.js for reliable matrix operations

Frequently Asked Questions

For a set of vectors to form a basis, they must be linearly independent and span the vector space. If the vectors are linearly dependent, they do not form a basis, and the change of basis matrix cannot be computed (the basis matrix would be singular and not invertible). The calculator will detect this and show an error message.

Change of basis is defined for bases of the same vector space, so both bases must have the same number of vectors (equal to the dimension of the space). Therefore, the basis matrices will always be square matrices (n × n for an n-dimensional space).

If A is the matrix of a linear transformation T with respect to basis B, and P is the change of basis matrix from B to B', then the matrix of T with respect to basis B' is P⁻¹AP. This is called a similarity transformation. Two matrices are similar if they represent the same linear transformation in different bases.

Geometrically, changing basis corresponds to describing the same vectors using a different coordinate system. For example, in ℝ², changing from the standard basis to a rotated basis corresponds to rotating the coordinate axes. The vectors themselves don't change, only their coordinates (numerical representation) change.

This calculator is designed for real vector spaces (ℝⁿ). For complex vector spaces (ℂⁿ), the principles are the same, but the calculator would need to handle complex numbers. Currently, it only accepts real number inputs.