Enter source and target bases for conversion. Instantly build the transition matrix and vector output. Review checks, exports, examples, and guided steps clearly today.
| Dimension | Source Basis B | Target Basis C | Vector in B | Transition Matrix C⁻¹B | Vector in C |
|---|---|---|---|---|---|
| 3 | [(1,0,0), (1,1,0), (0,1,1)] | [(1,1,0), (0,1,0), (0,0,1)] | (2,-1,3) | [[1,1,0],[-1,0,1],[0,1,1]] | (1,1,3) |
1. Build the source basis matrix: B = [b₁ b₂ ... bₙ].
2. Build the target basis matrix: C = [c₁ c₂ ... cₙ].
3. The change of coordinates matrix from B to C is PC←B = C⁻¹B.
4. If a vector is given in source coordinates, then [v]C = PC←B[v]B.
5. The standard vector is found by x = B[v]B.
6. A valid basis must be invertible, so det(B) ≠ 0 and det(C) ≠ 0.
This change of coordinates matrix calculator helps you convert vector coordinates from one basis to another. It is useful in linear algebra, matrix analysis, geometry, and advanced mathematics courses. Many students understand vectors in standard coordinates but struggle with custom bases. This tool removes that confusion. You enter a source basis, a target basis, and a vector written in the source basis. The calculator then builds the transition matrix and shows the converted coordinates.
A change of coordinates matrix is not just a random matrix. It connects two valid bases of the same vector space. The calculator forms the source basis matrix B and the target basis matrix C. It then computes C inverse times B. That product gives the matrix that changes coordinates from basis B to basis C. The page also checks determinants. This matters because a basis matrix must be invertible. If a determinant is zero, the entered vectors do not form a valid basis.
The result section shows more than one output. It gives the transition matrix, its reverse form, the original vector coordinates, and the final coordinates. It also reconstructs the standard vector. This lets you verify the answer quickly. You can compare the geometric object in different coordinate systems without changing the actual vector itself. That is the key idea in coordinate transformation, basis conversion, and vector representation.
This calculator is practical for homework checks, classroom demos, and self-study. It supports 2 by 2, 3 by 3, and 4 by 4 cases. That makes it flexible for common textbook exercises. The example table, export tools, and concise formulas also save time. If you need a reliable change of basis calculator, transition matrix finder, or coordinate conversion helper, this page is built for that purpose. It gives clear outputs with minimal clutter and simple workflow.
It converts coordinate vectors from one basis into another basis. The actual geometric vector stays the same. Only its coordinate description changes.
A basis must contain linearly independent vectors. That means its matrix must have a nonzero determinant. Without invertibility, coordinate conversion is not valid.
Use the transition matrix PC←B = C⁻¹B. Then multiply it by the vector coordinates in basis B to get the coordinates in basis C.
Yes. You can select dimension 2, 3, or 4. The input tables update automatically for the chosen matrix size.
The calculator stops and shows an error. A zero determinant means the entered vectors do not form a valid basis, so the transformation cannot be completed.
The standard vector helps verify the conversion. It confirms that both coordinate systems describe the same underlying vector in the vector space.
Yes. You can download a CSV summary of the inputs and outputs. You can also open a print-ready report and save it as a PDF.
Yes. It is helpful for lessons on basis change, transition matrices, vector spaces, and coordinate representations. The outputs are organized for quick explanation.