A set of vectors{v

1

,v

2

,,v

n

}

in a vector spaceV

is calledlinearly dependentif we can find scalarsr

1

,r

2

,,r

n

,not all zero, such that

r

1

v

1

+r

2

v

2

++r

n

v

n

=0

where the right hand side is the zero vector.

To demonstrate that a given set of vectors is linearly dependent, it is a good idea to use the definition and row-reduction to explicitlyfind scalars that demonstrate the linear relation.

If our investigation leads us to conclude that theonly wayto arrange a linear relation with right hand side being0

is to use

r

1

=r

2

==r

n

=0

A set of vectors {v1 ,VQ, - - - , v11} in a vector space V is calied linearly dependent if we can nd scalars 1'1 ,1'2, - - - ,rn, not ali zero, such that a Tlvl +r2v2 +---+rnvn : 0 where the right hand side is the zero vector. To demonstrate that a given set of vectors is iinearly dependent, it is a good idea to use the denition and rowereduction to explicitly nd scalars that demonstrate the linear reiation, If our investigation leads us to conclude that the only way: to arrange a linear relation with right hand side being 0 is to use 1'] ='l"2=---='l'n,=1 then the set cannot be linearly dependent (i.e the set is a linearly independent set). Let's show that 1 1 7 U: 4 ,V= 73 ,w: 77 1 72 2 is actually a linearly dependent set by nding scalars i", .9 and t (not all zero) such that ru + 3v i tw : 0. For example [Ti-Bit] =| In @, Using this we can express one of the vectors as a linear combination of the others, for example iin a