a set of vectors from a vector space is said to be linearly dependant if there exists a non-trivial (not all zeros) set of scalars , such that: thus, a set of vectors is linearly dependent if and only if one of them is zero or a linear combination of the others.
each of the vectors in a linearly dependant set is linearly dependant on the set
a set of vectors is just a subset of a vector space while a sequence of vectors is a map (can also be written as a infinite tuple). A set does not care about ordering or enumerating elements multiple times in contrast to a sequence.
given latex_all('v_1=', v1, ',v_2=', v2, ',v_3=', v3)
check whether these vectors are linearly dependant
we need to find such that , and the first combination that comes to mind is , so these vectors are linearly dependant
is a linearly dependant set if and only if
first we prove being linearly dependant means there exists such that: we multiply both sides of this equation by
second we prove we need to find such that and that can just be which would give us , therefore is linearly dependant