LP的线性独立性测试

Question

We have a set of vectors P1,...,Pk. Each vector has n dimension. These vectors are linearly independent if the only solution to the following problem is lambda(i) = 0 for each 0 <= i <= k:

lambda(1)P1 + lambda(2)P2 + ... + lambda(k)Pk = 0;

where lambda(i) is a real number. More rigorous formulation is available in https://en.wikipedia.org/wiki/Linear_independence

I am dealing with modeling this problem with an LP for a long while and so far no answer achieved. Would you please help me with this? Thanks.

Answer 1

Suppose you are dealing with a vector space of dimension n . If there are more vectors P1,...,Pk , then the input is not linearly independet; so suppose that k<=n . The goal is to determine the dimension of the linear hull of the input. Solve n linear programs of the form

max e_{i}x s. t. Ax = e_{i}

for each i in {1,...,n} where e_{i} denotes the i -th unit vector. The number of solvable linear programs should yield the dimension of the input's linear hull, which means that these number equals k if and only if the input is linearly independent.