Linear independence and rank
Alan G. Hamilton · Cambridge University Press eBooks · 1987
Examples 6.1 illustrate what is meant by linear dependence of a list of vectors. More formally: given a list of vectors v 1 ,…, v k of the same size (i.e. all are p × 1 matrices for the same p ), a linear combination of these vectors is a sum of multiples of them, i.e. x 1 v 1 + x 2 v 2 + … + x k v k where x 1 , …, x k are any numbers. A list of vectors is said to be linearly dependent (abbreviated to LD) if there is some non-trivial linear combination of them which is equal to the zero vector. Of course, in a trivial way, we can always obtain the zero vector by taking all of the coefficients x 1 , …, x k to be 0. A non-trivial linear combination is one in which at least one of the coefficients is non-zero. A list of vectors of the same size which is not linearly dependent is said to be linearly independent (abbreviated to LI). Example 6.2 deals with the case of a list of two 2-vectors. A list of two non-zero 2-vectors is LD if and only if each is a multiple of the other. Example 6.3 deals with a list of three 2-vectors. Such a list is always LD. Why? Because a certain set of simultaneous equations must have a solution of a certain kind.