Linear Independence Calculator – Formula and Process

The Linear Independence Calculator determines whether a given set of vectors are linearly independent or dependent.

Formula

Vectors 𝑣1,𝑣2,…,𝑣𝑛 are said to be linearly independent if:

has only the trivial solution:

Process

1. Form a Matrix

   Place the given vectors as columns in a matrix 𝐴=[𝑣1 𝑣2 … 𝑣𝑛].

2. Row Reduction (RREF)

   Apply Gaussian elimination to convert 𝐴 into its Reduced Row Echelon Form (RREF).

   • Each pivot (leading 1) represents a linearly independent column.
   • Zero rows indicate dependent relationships.
3. Rank Test
   • If the rank of 𝐴 equals the number of vectors, the vectors are independent.
   • If the rank is less than the number of vectors, they are dependent.
4. Determinant Method (for square matrices only)
   • If det⁡(𝐴)≠0 → Independent
   • If det(A)=0 → Dependent