Question

What does it mean when it says b is a non-trivial linear combination of the columns of matrix A? What is trivial and non-trivial in this context?

Answer 1

http://www.cliffsnotes.com/study_guide/Linear-Independence.topicArticleId-20807,articleId-20789.html
An alternative—but entirely equivalent and often simpler—definition of linear independence reads as follows. A collection of vectors v1, v2, …, v r from R n is linearly independent if the only scalars that satisfy :

k1v1+k1v2+k3v3+...knvn=0

are k1 = k2 = ⃛ = kn = 0. This is called the trivial linear combination. If, on the other hand, there exists a nontrivial linear combination that gives the zero vector, then the vectors are dependent.

see link for more explanations

Answer 2

So I can say that trivial merely means that the column vectors of a matrix A are all independent while non-trivial means the column vectors of matrix A are dependent?

Answer 3

yep. that's how im reading it

Answer 4

Thanks a lot! :)