Change of Basis Formula and Convention
Basis matrices
B = [b₁ … bₙ], C = [c₁ … cₙ]The ordered basis vectors are columns written in the same ambient coordinate system, usually standard coordinates.
B-coordinates to C-coordinates
[v]ᴄ = C⁻¹B[v]ᴮTherefore the displayed forward change of basis matrix is Pᴄ←ᴮ = C⁻¹B.
C-coordinates to B-coordinates
[v]ᴮ = B⁻¹C[v]ᴄThe reverse matrix is Pᴮ←ᴄ = B⁻¹C, the inverse of the forward matrix.
10⁻⁴⁰ through 10⁴⁰ in magnitude. Display precision does not round intermediate values.How to Use the Calculator
- Choose the dimension of the vector space.
- Enter the old ordered basis B and new ordered basis C. Put one basis vector in each column, with components down the rows.
- Optionally enter all coordinates of a vector relative to B.
- Select Calculate change of basis. Read
PC←Bfor the forward conversion andPB←Cfor the reverse conversion. - Use the residuals to check the floating-point calculation, then copy or download the result if needed.
Worked 2D Example
The default example uses B = [(1,1), (1,−1)], C = [(1,1), (0,2)], and [v]B = (3,−1). The basis vectors are columns, so:
Forward matrix
Pᴄ←ᴮ = [[1, 1], [0, −1]]This converts the vector to [v]C = (2,1).
Standard-coordinate check
B(3,−1) = (2,4) = C(2,1)Both coordinate columns describe the same vector in the ambient coordinate system.
Ordered Bases, Invertibility, and Numerical Limits
A coordinate column records the coefficients of a vector relative to an ordered basis. Because order determines which coefficient multiplies which basis vector, swapping two columns changes the conversion matrix. Both entered matrices must be square and nonsingular: if their columns are dependent, at least one vector has no unique coordinate column.
The calculator verifies CPC←B ≈ B and that the two conversion matrices multiply to the identity. Small nonzero residuals are normal in floating-point arithmetic. Large condition estimates mean rounding or small input changes may strongly affect the result.
Change of Basis Matrix FAQs
What is a change of basis matrix?
It converts the coordinate column of the same vector between two ordered bases. The columns of PC←B are the vectors of basis B expressed in basis C.
Why is the formula C⁻¹B rather than B⁻¹C?
Starting from v = B[v]B = C[v]C, multiply by C⁻¹ to get [v]C = C⁻¹B[v]B. The order reverses when converting in the other direction.
Does the order of basis vectors matter?
Yes. Coordinates belong to an ordered basis. Reordering columns changes which coefficient belongs to each basis vector.
What if one entered basis is singular?
Its columns are linearly dependent, so they do not form a basis and unique coordinate conversion is impossible. Check for duplicate, zero, or dependent basis vectors.
Are the forward and reverse matrices inverses?
Yes. (C⁻¹B)⁻¹ = B⁻¹C. The mutual-inverse residual checks this numerically.
Can I enter basis vectors as rows?
No. This calculator follows the common column-vector convention: place each basis vector in one column. If your source lists vectors as rows, transpose that arrangement first.
Are my entries private?
Yes. The page calculates locally and does not transmit or store your basis or vector entries. Copying and downloads are also created in your browser.
Calculation Notes
Last reviewed: August 5, 2026 by the Starlight Tools editorial team.
The coordinate-map definition and column-basis convention were checked against MIT OpenCourseWare’s change of basis materials. Numerical output is appropriate for study and routine calculation; use exact or higher-precision software when a proof or precision-sensitive result is required.