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