Linear Independance Calculator

u = ( a0 a1 )

v = ( b0 b1 )

w = ( c0 c1 )

t = ( d0 d1 )

u

v

w

t

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:

Linear Independance Formula

has only the trivial solution:

Linear Independance Formula

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)β‰ 0 β†’ Independent
    • If det(A)=0 β†’ Dependent