information:classification
Alternative method using determinants

An alternative method relies on the fact that n {\displaystyle n} vectors in R n {\displaystyle \mathbb {R} ^{n}} are linearly independent if and only if the determinant of the matrix formed by taking the vectors as its columns is non-zero.

In this case, the matrix formed by the vectors is

We may write a linear combination of the columns as

We are interested in whether A Λ = 0 for some nonzero vector Λ. This depends on the determinant of A {\displaystyle A}, which is

Since the determinant is non-zero, the vectors (1, 1) {\displaystyle (1,1)} and (− 3, 2) {\displaystyle (-3,2)} are linearly independent.

Otherwise, suppose we have m {\displaystyle m} vectors of n {\displaystyle n} coordinates, with m < n. {\displaystyle m<n.} Then A is an n × m matrix and Λ is a column vector with m {\displaystyle m} entries, and we are again interested in A Λ = 0. As we saw previously, this is equivalent to a list of n {\displaystyle n} equations. Consider the first m {\displaystyle m} rows of A {\displaystyle A}, the first m {\displaystyle m} equations; any solution of the full list of equations must also be true of the reduced list. In fact, if ⟨ i,..., i ⟩ is any list of m {\displaystyle m} rows, then the equation must be true for those rows.

Furthermore, the reverse is true. That is, we can test whether the m {\displaystyle m} vectors are linearly dependent by testing whether

for all possible lists of m {\displaystyle m} rows. (In case m = n {\displaystyle m=n}, this requires only one determinant, as above. If m > n {\displaystyle m>n}, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available.