Enter Linear Operator Data
Use the matrix form below to define a linear operator A on Rn. Then test vector mapping, additivity, homogeneity, and key operator properties.
Example Data Table
This sample uses a 3 × 3 operator to show how the calculator transforms vectors and confirms linearity properties.
| Item | Values | Interpretation |
|---|---|---|
| Operator A | [[2, 1, 0], [0, 3, -1], [1, 0, 2]] | Defines the linear transformation in R3. |
| Vector x | [1, 2, 1] | Main input vector for transformation. |
| Vector y | [0, 1, 3] | Secondary vector for additivity testing. |
| Scalar c | 2 | Used for the homogeneity test. |
| T(x) | [4, 5, 3] | Image of x under the operator. |
| T(y) | [1, 0, 6] | Image of y under the operator. |
Formula Used
Primary mapping: T(x) = Ax
Additivity test: T(x + y) = T(x) + T(y)
Homogeneity test: T(cx) = cT(x)
Rank-nullity: rank(A) + nullity(A) = n
Determinant rule: det(A) ≠ 0 implies invertibility
Norms: ||A||F = √∑aij2, ||A||1 = max column sum, ||A||∞ = max row sum
A linear operator on a finite-dimensional space can be represented by a square matrix. Multiplying that matrix by a vector gives the transformed vector.
The determinant measures volume scaling and signals whether the operator is invertible. Rank shows the dimension of the image, while nullity shows the dimension of the kernel.
The additivity and homogeneity checks numerically confirm that the entered matrix behaves exactly like a linear operator should.
How to Use This Calculator
1. Select the operator dimension. The page supports square matrices from 2 × 2 up to 5 × 5.
2. Enter the matrix values for A. This matrix defines the operator.
3. Enter vectors x and y, then enter a scalar c for the linearity tests.
4. Click Calculate Operator to display the results above the form.
5. Review T(x), T(y), determinant, rank, nullity, norms, inverse status, and eigen information.
6. Use the CSV button for spreadsheet-friendly output and the PDF button for a printable summary.
7. Use the Plotly graph to compare input and output vector coordinates visually.
FAQs
1. What does a linear operator calculator do?
It applies a square matrix to input vectors and reports transformation results. It also measures determinant, rank, nullity, norms, invertibility, and other operator properties from the same matrix.
2. Why must the matrix be square?
A linear operator maps a vector space to itself, so the input and output dimensions match. That requirement is represented by a square matrix.
3. What does the determinant tell me?
The determinant shows whether the operator compresses volume to zero. A nonzero determinant means the matrix is invertible and the operator is both one-to-one and onto.
4. What is nullity in this calculator?
Nullity is the dimension of the kernel, which contains all vectors sent to zero. Higher nullity means more information is lost by the operator.
5. Why are additivity and homogeneity shown?
These are the defining properties of linearity. The calculator compares both sides numerically so users can see the matrix satisfy T(x + y) = T(x) + T(y) and T(cx) = cT(x).
6. What does the dominant eigenvalue mean?
It estimates the strongest long-run scaling effect of the operator under repeated application. The associated eigenvector points in the direction that becomes most influential.
7. When is the inverse matrix shown?
The inverse appears only when the determinant is nonzero and the matrix is numerically invertible. Singular operators do not have an inverse matrix.
8. What do the norms help measure?
Norms summarize operator size and amplification strength. They help compare transformations, estimate sensitivity, and understand how strongly vectors can be stretched.