How this calculation works
Both square basis matrices contain their vectors in standard coordinates. Multiplying by B maps source coordinates to the actual standard-coordinate vector. Multiplying by C inverse then expresses that same vector using the target basis. Both inputs must be genuine ordered bases.
P(C ← B) = C⁻¹ B; [v]C = P [v]BA worked example
For standard B, C = [1 1; 1 −1] and source coordinates (3,1), the target coordinates are (2,1), since 2(1,1) + 1(1,−1) = (3,1).
Good to know
Exact rational matrices up to 6 × 6. Entries are bounded to ±1,000,000, integer fraction denominators to 1,000,000 and decimal input to 6 places. Intermediate fractions have a size limit. No complex entries, symbolic expressions, measurement uncertainty or numerical-conditioning analysis. Ordered vectors are entered as matrix columns.
Reference: Ohio State University: change of basis (opens in a new tab) ↗
The vector stays fixed while its coordinates change
The polynomial Jacobian calculator handles a nonlinear map locally; its derivative matrix can change with the input point rather than remaining one fixed basis conversion.
Coordinates describe how much of each ordered basis vector is needed. In the standard plane basis, (3,1) means three horizontal units and one vertical unit. In the basis (1,1), (1,−1), that same vector has coordinates (2,1). The point has not moved; the reference directions have changed.
Put basis vectors in columns, not rows. The first source coordinate multiplies column 1 of B, the second multiplies column 2, and so on. The order matters: exchanging two basis columns also exchanges the corresponding coordinate roles.
This tool accepts two bases of the full coordinate space. A dependent list cannot describe every vector uniquely, so it is rejected. Coordinates within a lower-dimensional embedded subspace require a different setup and are not silently treated as a square basis.
Check the direction by reconstructing the vector
For a transition from B coordinates to C coordinates, P = C⁻¹ B. Reversing the direction gives B⁻¹ C instead. A label such as “the change matrix” is incomplete unless the direction and coordinate conventions are stated.
Verify a result by multiplying B by the source coordinates and C by the returned target coordinates. Both products must give the same standard-coordinate vector. The result panel includes the standard coordinates so you can make this check directly.
Changing coordinates differs from applying a physical rotation or another linear operator. It is also different from changing the matrix representation of an operator, which involves a basis conversion on both sides. Use the operator-basis tool for that latter task.
A few questions, answered
Why must both matrices be invertible?
Their columns must each form an ordered basis. Invertibility guarantees unique coordinates for every vector in the full space.
What are the input vector entries?
They are coordinates in the source basis B. They are standard coordinates only when B is the standard identity basis.
Do the bases need to be orthogonal?
No. Any invertible real rational basis matrices within the input bounds are supported.