Linear Independence Calculator
Determine if a set of vectors is linearly independent using the determinant method.
Select Vector Space Dimension
Number of Vectors
How Linear Independence Works
A set of vectors {v₁, v₂, ..., vₙ} is linearly independent if the only solution to:
\[ c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n = \mathbf{0} \]
is c₁ = c₂ = ... = cₙ = 0.
Methods to check linear independence:
- Determinant Method:
- For n vectors in ℝⁿ, form a matrix with the vectors as columns
- Calculate the determinant
- If det ≠ 0, vectors are linearly independent
- Row Reduction Method:
- Form a matrix with the vectors as rows
- Reduce to row echelon form
- If no zero rows, vectors are linearly independent
Properties:
- Any subset of linearly independent vectors is also linearly independent
- Any superset of linearly dependent vectors is also linearly dependent
- In ℝⁿ, any set of more than n vectors must be linearly dependent
- In ℝⁿ, any set of n linearly independent vectors forms a basis
Applications:
- Basis determination
- Dimension calculation
- Span analysis
- Solution space analysis
- Eigenvalue problems