A set of vectors {vj}j=1N{vj}j=1N is said to be *linearly dependent*
is there exists a set of scalars α1,...,αNα1,...,αN (not all 0) such that

∑
j
=
1
N
α
j
v
j
=
0
.
∑
j
=
1
N
α
j
v
j
=
0
.

(1)
Likewise if ∑j=1Nαjvj=0∑j=1Nαjvj=0 only when αj=0αj=0∀j∀j,
then {vj}j=1N{vj}j=1N is said to be *linearly independent*.

v
1
=
2
1
0
,
v
2
=
1
1
0
,
v
3
=
1
2
0
.
v
1
=
2
1
0
,
v
2
=
1
1
0
,
v
3
=
1
2
0
.

(2)
Find α1,α2,α3α1,α2,α3 such that α1v1+α2v2+α3v3=0α1v1+α2v2+α3v3=0. [α1=1,α2=-3,α3=1α1=1,α2=-3,α3=1.] Note that any two vectors are linearly independent.

Note that if a set of vectors {vj}j=1N{vj}j=1N are linearly dependent
then we can remove vectors from the set without changing the span of the set.