Calculator Input
Example Data Table
| Matrix | Vector orientation | Rank | Vector count | Result | Reason |
|---|---|---|---|---|---|
| [[1,2,3],[2,4,6],[3,6,9]] | Columns | 1 | 3 | Dependent | Only one pivot column exists. |
| [[1,0,0],[0,1,0],[0,0,1]] | Columns | 3 | 3 | Independent | Every column is a pivot column. |
| [[1,2],[3,4],[5,6]] | Rows | 2 | 3 | Dependent | Three row vectors lie in two dimensions. |
Formula Used
The calculator uses rank tests, row reduction, determinant checks, and nullspace relations.
Column test: columns are independent when rank(A) equals the number of columns.
Row test: rows are independent when rank(A) equals the number of rows.
Dependence rule: vectors are dependent when rank(A) is less than the tested vector count.
Nullity: nullity(A) = number of columns - rank(A).
Square shortcut: a square matrix with nonzero determinant has independent columns.
How to Use This Calculator
- Enter the matrix row count and column count.
- Select whether your vectors are stored as rows or columns.
- Type every matrix value into the input table.
- Set a small zero tolerance for decimal roundoff.
- Press Calculate to view rank, pivots, RREF, and dependence.
- Use CSV or PDF buttons to save the report.
Linear Dependence Matrix Guide
What Linear Dependence Means
Linear dependence means one vector can be made from others. A matrix gives a fast way to test that idea. Place the vectors in rows or columns. Then compare rank with the number of tested vectors. If rank is smaller, the set is dependent. If rank matches, the set is independent.
Why Rank Matters
Rank counts independent directions in a matrix. Row reduction exposes those directions. Pivot positions show the leading variables. Free variables show possible dependency relations. A free variable means a nonzero coefficient solution exists. That solution proves dependence through a linear combination.
Advanced Matrix Checks
This calculator handles rectangular and square matrices. It supports row vectors and column vectors. It also reports nullity for the equation A x equals zero. For square matrices, the determinant gives a useful shortcut. A nonzero determinant means full rank. A zero determinant means the columns are dependent.
Using Tolerance Carefully
Decimal data can create tiny roundoff errors. The tolerance value treats very small numbers as zero. Use a smaller tolerance for exact integer work. Use a larger tolerance for measured decimal data. A poor tolerance may hide pivots or create false pivots. Always review the reduced matrix before final conclusions.
Reading the Output
The result panel gives a direct dependence decision. Rank tells how many independent directions remain. Pivot columns identify essential columns in reduction. Nullity counts free choices in A x equals zero. The dependency basis gives coefficient patterns. These coefficients can form equations among your vectors.
Common Study Uses
Students use this tool for linear algebra homework. Teachers use it to prepare examples. Engineers use rank checks in systems and models. Data analysts use dependence tests before regression work. The export buttons help save clear records. The example table shows expected outcomes for common cases.
FAQs
What is linear dependence?
Linear dependence occurs when at least one vector can be written as a combination of other vectors in the same set.
How does this calculator test dependence?
It reduces the matrix, finds rank, checks pivots, and compares rank with the number of selected row or column vectors.
Should I choose rows or columns?
Choose columns when each vector is a column. Choose rows when each vector is written across one matrix row.
What does rank mean here?
Rank is the number of independent directions found in the matrix after row reduction. More pivots mean higher rank.
What does nullity mean?
Nullity is the number of free variables in A x equals zero. It equals the column count minus matrix rank.
Can a rectangular matrix have dependent vectors?
Yes. Rectangular matrices can have dependent rows or columns. The decision depends on rank and the selected vector orientation.
Why is the determinant sometimes unavailable?
The determinant is only defined for square matrices. Rectangular matrices need rank and row reduction instead.
What tolerance should I use?
Use a very small tolerance for exact values. Use a larger tolerance when inputs come from measurements or rounded decimals.