Understand the subject
What is linear independence?
Vectors are independent when none can be built from the others.
Use it to check whether every listed direction contributes new information.
See the structure
What the calculation is doing
v₁v₂independent directions span a plane
Worked interpretation
Read the result in context
independent ⇔ rank equals number of columns
Interpret with care
Important boundary
One nontrivial combination equalling zero proves dependence.
Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.
Quick guide
How to use this calculator
- Set the matrix dimensions required by the task.
- Fill every visible entry and any operation-specific control.
- Read the result type, factors, classification, and disclosed limitations.
Calculation method
Apply the stated linear-algebra contract
Columns are independent exactly when rank(A) equals the number of columns.
Undefined, singular, dimensionally incompatible, dependent, or unsupported inputs receive an explicit message instead of a fabricated numeric result.
Worked example
Example interpretation
A nontrivial null-space direction proves dependence.
Columns are independent exactly when rank(A) equals the number of columns.
Supported inputs
Precision and limits
Matrix size
Input matrices are limited to 1–6 rows and columns; operation-specific square, rank, or compatibility rules still apply.
Entry precision
Enter signed finite decimals with at most 30 digits and 15 decimal places.
Exactness
Finite decimals become reduced rational numbers and intermediate structural calculations are not rounded.
