A set of vectors \(\{v_1, v_2, ..., v_k\}\) is said to be linearly independent if the only solution to the equation:

\(c_1v_1 + c_2v_2 + ... + c_kv_k = 0\)

is the trivial solution, where all scalars \(c_1, c_2, ..., c_k\) are zero.

If there is at least one non-zero scalar that satisfies the equation, the set is linearly dependent.

How it's Solved:

To check for linear independence, we can place the vectors as columns in a matrix \(A\). The set is linearly independent if and only if the equation \(Ax = 0\) has only the trivial solution (\(x=0\)).

This calculator uses Gauss-Jordan elimination to reduce the matrix to its row echelon form. If every column has a pivot (a leading 1), there are no free variables, and the set is linearly independent. Otherwise, it is linearly dependent.