Linear Independence - 1.7
1. Definition:
A set of p vectors v 1 , v 2 , ! , v p
in R n is said to be linearly independent if the vector equation
x1v1 " x2v2 " ! " xpvp # 0
has only the trivial solution. A set of p vectors v 1 , v 2 , ! , v p in R n is said to be linearly dependent
if there exist constants: c 1 , ! , c p , not all zeros, such that
c 1 v 1 " c 2 v 2 " ! " c p v p # 0.
The equation c 1 v 1 " c 2 v 2 " ! " c p v p # 0 is called a linear dependence relation among vectors
v1, v2, ! , vp.
Remarks:
a. When p # 2, if v 1 , v 2
is linearly dependent then there exist c 1 and c 2 , not all zeros such that
c 1 v 1 " c 2 v 2 # 0.
Assume that c 1 ! 0, then
v 1 # " cc 21 v 2 ,
v 1 is a linear combination of v 2 . So, if v 1 is not a linear combination of v 2 , then v 1 , v 2 is linearly
independent.
b. Similarly, if v 1 is a linear combination of v 2 , ! , v p , then v 1 , v 2 , ! , v p is linearly dependent.
On the other hand, if v 1 , v 2 , ! , v p is linearly independent, then none of these vectors is a linear
combination of the others.
c. When v 1 , v 2 , ! , v p are columns of a matrix A, then the linear dependence relation
x1v1 " x2v2 " ! " xpvp # 0
among columns of A corresponds to a nontrivial solution Ax # 0. Hence, columns of A are linearly
independent if and only if the equation Ax # 0 has only trivial solution (unique solution).
1
Example Let v 1 #
2
4
, v2 #
5
3
a.
b.
c.
a.
, v3 #
6
1
.
0
Determine if the set v 1 , v 2 is linearly independent.
Determine if the set v 1 , v 2 , v 3 is linearly independent.
If possible, find a linear dependence relation among v 1 , v 2 , v 3 .
Because v 1 ! cv 2 , v 1 , v 2 is linearly independent.
1 4 2
b.
2
v1 v2 v3
#
""2#R 1 " R 2 $ R 2
3 6 0
""3#R 1 " R 3 $ R 3
""2#R 2 " R 3 $ R 3
%
1 4
2
0 "3 "3
%
2 5 1
1 4
0 "6 "6
2
0 "3 "3
0 0
0
Because there are c 1 and c 2 (here c 2 # 1 and c 1 # "2# such that v 3 # c 1 v 1 " c 2 v 2 , a linear
combination of v 1 and v 2 , v 1 , v 2 , v 3 is linearly dependent.
c. v 3 # ""2#v 1 " "1#v 2 .
0 1 4
Example Determine if the columns of the matrix A #
1 2 "1
are linearly independent. Does the
5 8 0
system Ax # b have a unique solution for any vector b in R 3 ?
0 1 4
A#
1 2 "1
R1 & R2
%
5 8 0
1 2 "1
0 1 4
""5#R 1 " R 3 & R 3
%
5 8 0
1 2
"1
0 1
4
0 "2 5
1 2 "1
2R 2 " R 3 & R 3
0 1 4
%
0 0 13
Because Ax # 0 has only trivial solution, columns of A are linearly independent. The system Ax # b have
a unique solution for any vector b in R 3 since columns of A are linearly independent.
2. Properties:
a. Let v 1 , v 2 , ! , v p be vectors in R n . If p $ n, then v 1 , v 2 , ! , v p is linearly dependent.
Proof: Let A # v 1 v 2 ! v p . Then A is an n % p matrix. Because p $ n, the reduced matrix of A
can have n leading elements at most. Hence, the system Ax # 0 has at least one nonzero solution and
v 1 , v 2 , ! , v p is linearly dependent.
b. Let v 1 , v 2 , ! , v p be vectors in R n . If one of these vectors is a zero vector, then v 1 , v 2 , ! , v p
is linearly dependent.
Proof: Let v 1 # 0 and x 1 ! 0. Then
x 1 v 1 " x 2 v 2 " ! " x p v p # 0 ' x 2 v 2 " ! " x p v p # "x 1 v 1 ' v 1 # " x11 "x 2 v 2 " ! " x p v p #.
Hence, v 1 is a linear combination of v 2 , ! , v p and v 1 , v 2 , ! , v p is linearly dependent.
Example Let v 1 #
2
1
, v2 #
4
"1
, v3 #
"2
2
.
a. Determine if the set v 1 , v 2 , v 3 is linearly independent.
b. Give two linear independent vectors among v 1 , v 2 , v 3 if possible.
a. Because p # 3 $ n # 2,
b.
v1 v2 v3
v1, v2
#
2 4
v1, v2, v3
"2
1 "1 2
is linearly dependent.
" 12 R 1 " R 2 $ R 2
%
is linearly independent.
Example Determine if the given set is linearly dependent.
2 4
"2
0 "3 3
1
a.
b.
7
2
,
,
0
1
6
9
5
2
0
1
3
,
0
"2
4
6
10
,
0
5
c.
3
4
,
1
8
1
8
3
,
"6
"9
15
a. Because p # 4 $ n # 3, v 1 , v 2 , v 3 , v 4 is linearly dependent.
b. Because v 2 # 0, v 1 , v 2 , v 3 is linearly dependent.
c. Because v 1 ! cv 2 , v 1 , v 2 is linearly independent.