Last updated: August 1, 2026
Linear Independence Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
Enter one vector per row to form a matrix, then compare the matrix rank with the number of vectors. If rank and vector count match, the set is linearly independent. If the rank is smaller, the set is dependent because at least one vector can be built from the others.
Enter one vector per row to compare matrix rank with the number of vectors and decide whether the set is independent or dependent.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
A set of vectors is linearly independent exactly when the rank of the matrix of vectors equals the number of vectors.
Where:
- V=matrix of input vectors
- rank(V)=pivot count after elimination
- m=number of input vectors
Worked Examples
Independent pair in R²
The standard basis vectors are independent.
- 1Reduce the matrix.
- 2Keep both pivots.
- 3Compare rank with vector count.
Dependent pair
One vector is a multiple of the other.
- 1Reduce the second row to zero.
- 2Rank drops below the vector count.
Three independent vectors in R³
Use the identity rows.
- 1Each row keeps its own pivot.
- 2Rank matches the number of vectors.
Too many vectors for the dimension
Three vectors in R² cannot all be independent.
- 1Notice the ambient dimension is only 2.
- 2Dependence is unavoidable.
Introduction
Check whether a set of vectors is linearly independent by comparing rank with the number of vectors.
Overview
Check whether a set of vectors is linearly independent by comparing rank with the number of vectors.
Independence classification
Basis selection
Span analysis
Redundancy checks
Input format
Enter one vector per row, such as 1,0;0,1.
Separate rows with semicolons
Separate entries with commas or spaces
Brackets are optional
Keep row lengths consistent
Formula and method
A set of vectors is linearly independent exactly when the rank of the matrix of vectors equals the number of vectors.
- V matters:
matrix of input vectors.
- Rank(V) matters:
pivot count after elimination.
- M matters:
number of input vectors.
Validation rules
The calculator uses deterministic validation before it computes the final result.
The input must form a rectangular matrix
All entries must be finite
Empty matrices are rejected
Rank is computed by row-echelon reduction
Interpreting the outputs
Use the main result first, then the secondary fields to verify the structure and meaning of the answer.
Independent means rank equals vector count
Dependent means at least one vector adds no new direction
Rank also tells you the dimension of the span
Vector count reminds you how many rows were tested
Common use cases
These are typical reasons to use this calculator in study or applied work.
Basis selection
Span analysis
Redundancy checks
Preparation for rank and null space problems
FAQs
Should I enter vectors as rows or columns?
Enter them as rows; the independence result is unchanged because rank is unchanged by transpose.
Why can three vectors in R² not all be independent?
Because the rank in R² can never exceed 2.
What is a pivot?
A pivot is a leading nonzero entry after elimination, and the number of pivots equals the rank.
Can I use decimals?
Yes. Any finite real entries are supported.
What does dependent mean?
It means at least one vector can be written as a combination of the others.
What if the vectors have different lengths?
The calculator rejects the input because it is not a valid matrix.