University of Regina MATH 122 - Linear Algebra I
MATH 122 - Linear Algebra I
§5.2. Linear independence
Martin Frankland
September 19, 2022
Definition 1. Vectors ⃗v1 , . . . , ⃗vk in Rn are called linearly independent if the only linear
combination that yields the zero vector
c1⃗v1 + · · · + ck⃗vk = ⃗0 (1)
is the trivial combination c1 = c2 = . . . = ck = 0. Otherwise the vectors are called linearly
dependent. A nontrivial linear combination of the form (1) is called a linear dependence
relation among the vectors ⃗v1 , . . . , ⃗vk .
Example 2. Are the given vectors in R3 linearly independent? If so, prove it; if not, find a
linear dependence relation among the vectors.
4 1 −2
(a) ⃗v1 = 1 , ⃗v2 = 3 , and ⃗v3 = 1 .
−1 1 2
Solution. The coefficients c1 , c2 , c3 in the equation
c1⃗v1 + c2⃗v2 + c3⃗v3 = ⃗0
are the unknowns in the homogeneous linear system with the ⃗vi as columns:
4 1 −2 0 1 3 1 0 1 3 1 0
R1 ↔R2 R2 −4R1 R2 +3R3
⃗v1 ⃗v2 ⃗v3 ⃗0 = 1 3 1 0 ∼ 4 1 −2 0 ∼
0 −11 −6 0 ∼
R3 +R1 “improve the pivot”
−1 1 2 0 −1 1 2 0 0 4 3 0
1 3 1 0 1 3 1 0
0 1 3 0 R3 −4R ∼ 2 0 1 3 0 .
0 4 3 0 0 0 −9 0
Since the matrix has rank 3, there is only the trivial solution c1 = c2 = c3 = 0. Hence the
vectors ⃗v1 , ⃗v2 , and ⃗v3 are linearly independent .
© 2022 Martin Frankland All Rights Reserved 1
MATH 122 - Linear Algebra I
§5.2. Linear independence
Martin Frankland
September 19, 2022
Definition 1. Vectors ⃗v1 , . . . , ⃗vk in Rn are called linearly independent if the only linear
combination that yields the zero vector
c1⃗v1 + · · · + ck⃗vk = ⃗0 (1)
is the trivial combination c1 = c2 = . . . = ck = 0. Otherwise the vectors are called linearly
dependent. A nontrivial linear combination of the form (1) is called a linear dependence
relation among the vectors ⃗v1 , . . . , ⃗vk .
Example 2. Are the given vectors in R3 linearly independent? If so, prove it; if not, find a
linear dependence relation among the vectors.
4 1 −2
(a) ⃗v1 = 1 , ⃗v2 = 3 , and ⃗v3 = 1 .
−1 1 2
Solution. The coefficients c1 , c2 , c3 in the equation
c1⃗v1 + c2⃗v2 + c3⃗v3 = ⃗0
are the unknowns in the homogeneous linear system with the ⃗vi as columns:
4 1 −2 0 1 3 1 0 1 3 1 0
R1 ↔R2 R2 −4R1 R2 +3R3
⃗v1 ⃗v2 ⃗v3 ⃗0 = 1 3 1 0 ∼ 4 1 −2 0 ∼
0 −11 −6 0 ∼
R3 +R1 “improve the pivot”
−1 1 2 0 −1 1 2 0 0 4 3 0
1 3 1 0 1 3 1 0
0 1 3 0 R3 −4R ∼ 2 0 1 3 0 .
0 4 3 0 0 0 −9 0
Since the matrix has rank 3, there is only the trivial solution c1 = c2 = c3 = 0. Hence the
vectors ⃗v1 , ⃗v2 , and ⃗v3 are linearly independent .
© 2022 Martin Frankland All Rights Reserved 1
