Linear Independence Calculator

A set of vectors is defined as linearly independent if no vector in the set can be expressed as a linear combination of the others. Mathematically, the vector equation c1v1 + c2v2 + ... + cnvn = 0 has only the trivial solution where every scalar c equals zero. To test a set of vectors computationally, we arrange them as columns or rows inside a matrix and calculate its rank. If the matrix rank equals the number of vectors, the set is linearly independent. If the rank is strictly less than the number of vectors, at least one vector is redundant, making the set linearly dependent.

Let us test whether the following three 3-dimensional vectors are linearly independent: v1 = [1, 2, 1], v2 = [0, 1, 3], and v3 = [2, 7, 11]. First, we construct a 3x3 matrix using these vectors as columns: Matrix = [[1, 0, 2], [2, 1, 7], [1, 3, 11]]. Next, we apply Gaussian elimination to find the row echelon form. Subtracting twice the first row from the second row and once from the third row gives: [[1, 0, 2], [0, 1, 3], [0, 3, 9]]. Finally, subtracting three times the second row from the third row yields: [[1, 0, 2], [0, 1, 3], [0, 0, 0]]. Since the row echelon form contains only two non-zero rows, the rank of this matrix is 2. Because the rank (2) is less than the number of vectors (3), the original set of vectors is linearly dependent.

When testing vectors, always ensure they share the same dimension; comparing vectors of different lengths is mathematically invalid. Keep in mind that any set containing a zero vector is automatically linearly dependent because the zero vector can be generated using a non-zero scalar multiplied by any other vector. Additionally, if the number of vectors in a set exceeds the dimension of the space they inhabit, the set is guaranteed to be linearly dependent.