5.3 Linear Independence
Linear Independence
 Definition:



If S = {v1,v2,...,vr} is a non empty set of vectors
then the vector equation k1v1+k2v2+....+kr vr =
0 has at least one solution, k1=0, k2=0, ...,
kr=0.
If this is the only solution, then S is called a
linearly independent set.
If there are other solutions, then S is called a
linearly dependent set.
2
Linear Independence
 Example:



v1 = (2,-1,0,3), v2=(1,2,5,-1), v3=(7,-1,5,8)
S={v1, v2, v3} is linearly dependent since
3v1+v2 – v3 =0.
S = {p1=1-x, p2=5+3x-2x2, p3=1+3x-x2} is a
linearly dependent since 3p1-p2+2p3=0
S={i, j, k}, where i=(1,0,0), j=(0,1,0), k=(0,0,1),
is a linearly independent since 0i + 0j + 0k =
0;
3
Linear Independence
 Theorem 5.3.1: A set S with two or more
vectors is
a) Linearly dependent iff at least one of the
vectors in S is expressible as a linear
combination of the other vectors in S
b) Linearly independent iff no vectors in S is
expressible as a linear combination of the
other vector in S.
 Example:
V1 = (2, -1, 0, 3), V2 = (1, 2, 5, -1), V3 = (7, -1, 5, 8)
V1 = -⅓ V2 + ⅓ V3, V2 = -3V1+V3, V3 = -3V1+V2
4
Linear Independence
Theorem:
a) A finite set of vectors that contain the zero vectors is
linearly dependent
b) A set with exactly two vectors is linearly independent
iff neither vector is a scalar multiple of the other.
5
Geometric Interpretation of Linear
Independence
 In R2 or R3, a set of two vectors is linearly
independent iff the vectors do not lie on the
same line when they are placed with their
initial points at the origin.
6
Geometric Interpretation of Linear
Independence
 In R3, a set of three vectors is linearly
independent iff the vectors do not lie in the
same plane when they placed with their initial
points at the origin.
7
Geometric Interpretation of Linear
Independence
 Theorem 5.3.3: Let S={v1,v2,...,vr} be a set vectors in Rn. If r>n,
then S is linearly dependent.
Proof:
v 1  (v11 , v12 ,..., v1n )
v 2  (v21 , v22 ,..., v2 n )


v r  (vr1 , vr 2 ,..., vrn )
k1v1  k2 v 2  ...  kr vr  0
v 11k1  v 21k 2    v r1 k r  0
v 12 k1  v 22 k 2    v r 2 k r  0




v 1n k1  v 2n k 2    v rn k r  0
homoggeneous system of n
equations in the r unknowns
k1,...,kr. Since r>n, the system has
nontrivial solutions. Therefore, S is
a linearly dependent set.
8
Linear Independence of Functions
n 1
If f1  f1 ( x), f 2  f 2 ( x),..., f n  f n ( x) are n-1 times C ( , )
differentiable functions on interval (, ), then the determinant of
f 2 ( x) ... f n ( x) 
 f1 ( x)
 f ' ( x)

'
'
f 2 ( x) ... f n ( x) 
1

W ( x) 
 ...
...
...
... 
 ( n1)

( n 1)
( n 1)
 f 1 ( x) f 2 ( x) ... f n ( x) 
is called the Wronskian of f1 , f 2 ,..., f n
9
Linear Independence of Functions
 Theorem: If the functions f1, f2,..., fn have n-1
continuous derivatives on the interval (-~,~), and if
the Wronskian of these functions is not identically
zero on (-~,~), then these functions form a linearly
independent set of vectors in C(n-1)(-~,~).
 Example: Linearly Independent Set in C1(-~,~)
Show that f1=x and f2=sin x form a linearly
independent set of vectors in C1(-~,~).
x sin x
W ( x) 
 x cos x  sin x
1 cos x
The function does not have value 0 for all x in the
interval (-~,~), f1 & f2 form a linearly independent set
10
Linear Independence of Functions
 Example: Linearly Independent Set in C2(-~,~)
Show that f1=1, f2=ex, and f3=e2x form a linearly
independent set of vectors in C2(-~,~).
1
W ( x)  0
0
ex
ex
ex
e2 x
2e 2 x  2 e 3 x
4e 2 x
This function does not have value zero for all x in the
interval (-~,~), so f1, f2, and f3 form a linearly
independent set.
11