Linear independence
A set vectors, 
are said to be linearly independent if a linear combination of the vectors is zero if and only if all the coefficients of the vectors are zero.[1] In other words, the equation:
only holds when 
all equal 0. If this equation is true for any other values of , then the vectors are said to be "linearly dependent". If a set of n vectors, each with dimension n, are linearly independent, then they form a basis for that space.[2]
Testing for Linear Independence
The simplest way to test for linear independence is to try to express one vector as a linear combination of the others. Consider the vectors:
These are not linearly independent as the third vector can be written as a linear combination of the other two:
Another method is to combine the vectors into a square matrix and calculates its determinant.[3] If the vectors are linearly dependent, the determinant will be zero. The vectors above can be combined into a matrix as the columns of the matrix.
The order of the vectors in the matrix does not matter. The determinant can be found as:
As the determinant is not zero, the vectors are not linearly dependent. This matrix method only works when the dimension of the vectors and the number of vectors are the same.
