Calculator
Example Data Table
| Dimension | A | B | C | vB | Purpose |
|---|---|---|---|---|---|
| 2 | [2, 1] | [0, 3] | [1, 1] | [0, 1] | [1, 0] | [1, 1] | [2, -1] | Find [T]C←B and the transformed coordinates. |
| 3 | [1, 0, 2] | [0, 1, 1] | [2, 0, 1] | [1, 0, 0] | [1, 1, 0] | [0, 1, 1] | [1, 1, 0] | [0, 1, 1] | [1, 0, 1] | [3, 2, 1] | Check a three dimensional basis transformation. |
Formula Used
The calculator uses the coordinate matrix formula:
[T]C←B = C-1AB
Here, A is the standard matrix of the transformation. B is the domain basis matrix. C is the codomain basis matrix.
For a coordinate vector vB, the standard input vector is v = BvB.
The standard image is T(v) = Av. The target coordinates are [T(v)]C = C-1ABvB.
How to Use This Calculator
- Select the matrix dimension.
- Enter the standard transformation matrix A.
- Enter the domain basis matrix B, using basis vectors as columns.
- Enter the codomain basis matrix C, using target basis vectors as columns.
- Enter an optional coordinate vector vB.
- Choose decimal places.
- Press the calculate button.
- Download the CSV or PDF report after results appear.
What This Calculator Does
A coordinate matrix explains a linear transformation in chosen bases. The same transformation can look different when the basis changes. This calculator handles that work with direct matrix operations. It accepts a standard transformation matrix, a domain basis, a codomain basis, and an optional coordinate vector. It then builds the coordinate matrix that maps input coordinates to output coordinates.
Why Basis Choice Matters
In linear algebra, a vector is not only a list of numbers. The list depends on the basis used to describe it. A coordinate vector in one basis can represent the same geometric vector as another list in another basis. This is why basis conversion is important. It keeps the map consistent, even when the coordinate system changes.
Practical Matrix Checks
The calculator also checks whether each basis matrix is usable. A basis matrix must be square and invertible. If its determinant is zero, the columns do not form a valid basis. The rank value helps confirm this. Trace and determinant are included for quick review of the transformation matrix. These checks support homework, lectures, and study notes.
Interpreting the Result
The coordinate matrix sends a vector written in the domain basis into coordinates written in the codomain basis. If you enter vector coordinates, the page also shows the standard input vector, the standard image, and the final output coordinates. This lets you compare every stage of the transformation.
Good Uses
Use the tool for two dimensional and three dimensional examples. You can also test four dimensional class problems. It is useful for rotations, projections, shears, scalings, reflections, and abstract basis exercises. Because the calculator shows formulas and intermediate results, it helps users find entry mistakes and understand the process.
Exporting Work
After calculation, download a CSV file for spreadsheet review. You can also download a compact PDF report for sharing or printing. These reports are useful when you need to save several practice cases. Always review the entered matrices before submitting. A small misplaced value can change the entire coordinate matrix.
For stronger practice, compare the result with manual row reduction. Then verify each coordinate product step by step before using it in proofs, assignments, or design notes for better accuracy later.
FAQs
What is a coordinate matrix?
A coordinate matrix represents a linear transformation between chosen bases. It converts input coordinates in one basis into output coordinates in another basis.
What does matrix A mean?
Matrix A is the standard matrix of the linear transformation. It acts on vectors written in the usual standard coordinates.
What is matrix B?
Matrix B stores the domain basis vectors as columns. It converts domain coordinates into standard coordinates before applying the transformation.
What is matrix C?
Matrix C stores the codomain basis vectors as columns. Its inverse converts standard output vectors into target basis coordinates.
Why must B and C be invertible?
They must be invertible because their columns must form valid bases. A singular basis matrix cannot describe every vector uniquely.
Can I enter fractions?
Yes. You can enter values such as 1/2, -3/4, or 5/2. The calculator converts them into decimal values for computation.
What dimensions are supported?
The calculator supports 2 by 2, 3 by 3, and 4 by 4 matrices. These cover many common class examples.
What does the PDF export include?
The PDF export includes entered data, formulas, matrix checks, the coordinate matrix, and vector results when a vector is provided.