Calculator Inputs
Enter basis vectors as columns. The source basis is B. The target basis is C.
Formula Used
Let B be the source basis matrix. Its columns are the old basis vectors. Let C be the target basis matrix. Its columns are the new basis vectors.
The main transition matrix is:
P = C⁻¹B
If x is a coordinate vector in basis B, then the target coordinate vector is:
[v]C = P[v]B
If a standard linear map A is supplied, the matrix of that map from basis B to basis C is:
[T]C←B = C⁻¹AB
The calculator also checks CP = B. A small residual means the result is numerically consistent.
Example Data Table
| Case | Source Basis B | Target Basis C | Main Formula | Expected Use |
|---|---|---|---|---|
| Identity to skew basis | [[1,0],[0,1]] | [[1,1],[0,1]] | P = C⁻¹B | Convert standard coordinates to skew coordinates. |
| Three dimensional basis | Identity 3 × 3 | [[1,1,0],[0,1,1],[1,0,1]] | P = C⁻¹B | Study coordinate changes in space. |
| Linear map included | Any valid B | Any valid C | C⁻¹AB | Represent a transformation between two bases. |
| Repeated transition | Any valid B | Any valid C | Pⁿ | Review repeated coordinate movement. |
How to Use This Calculator
- Select the dimension of your basis matrices.
- Enter the source basis vectors as columns in matrix B.
- Enter the target basis vectors as columns in matrix C.
- Add a standard matrix A only when you want a transformed map.
- Enter a coordinate vector in basis B if needed.
- Choose decimal precision and matrix power.
- Press the calculate button.
- Review the result, graph, checks, CSV export, and PDF export.
Transition Matrix Guide
What a transition matrix means
A transition matrix changes coordinates from one basis to another. The vector itself does not move. Only its description changes. This idea is central in linear algebra. It appears in geometry, computer graphics, differential equations, quantum mechanics, and data modeling.
Why bases matter
A basis is a set of independent vectors that spans a space. In two dimensions, a basis has two vectors. In three dimensions, it has three vectors. When these vectors are placed as columns, they form a basis matrix. The determinant must not be zero. A zero determinant means the vectors are dependent. Then the inverse does not exist.
How the calculator works
This calculator places the source basis into matrix B. It places the target basis into matrix C. The transition matrix is found with C inverse times B. Each column of the answer tells how one source basis vector is written in the target basis. This gives a direct coordinate conversion rule.
Using a transformation matrix
Some problems also include a standard linear transformation. This calculator supports that option. If matrix A is entered, the calculator forms C inverse times A times B. That result represents the same transformation, but it starts with source basis coordinates and ends with target basis coordinates.
Checking the answer
The determinant values help confirm that the bases are valid. The residual check compares C times P with B. A very small residual means the transition matrix is correct within rounding limits. The condition estimate warns about unstable inputs. Large values may amplify small entry errors.
Practical use
Use the CSV export for spreadsheets and classroom records. Use the PDF export for reports or homework notes. The Plotly heatmap helps locate large positive and negative entries quickly. The powered matrix option is useful when repeated transitions need inspection. Always enter basis vectors as columns, not rows.
FAQs
1. What is a transition matrix?
A transition matrix converts coordinates from one basis to another. It does not change the actual vector. It only changes how that vector is described.
2. Which formula does this calculator use?
It uses P = C⁻¹B. Matrix B stores the source basis. Matrix C stores the target basis. The result converts B-coordinates into C-coordinates.
3. Should basis vectors be entered as rows or columns?
Enter basis vectors as columns. This is the standard setup for the formula P = C⁻¹B. Row entry will produce a different result.
4. Why does a singular matrix stop the calculation?
A singular basis matrix has no inverse. Its vectors are dependent. Such vectors cannot form a valid basis for coordinate conversion.
5. What does C⁻¹AB mean?
It represents a standard linear transformation A from source basis coordinates to target basis coordinates. Use it when a transformation is part of the problem.
6. What is the residual check?
The calculator checks whether C times P equals B. A small residual shows the transition matrix matches the original basis relation accurately.
7. Can this calculator handle fractions?
Yes. You can enter values like 1/2 or -3/4. The calculator converts them into decimal values for calculation and display.
8. What does the matrix power option do?
It raises the transition matrix to a chosen power. This helps inspect repeated coordinate changes or compare multi-step transition behavior.