MAT 102
LINEAR DEPENDENCE AND
INDEPENDENCE
DEFINITION
If 𝑆 = {𝑣1 , 𝑣2 , … , 𝑣𝑛 } is a nonempty set of vector, then the vector
equation
𝑥1 𝑣1 + 𝑥2 𝑣2 + ⋯ + 𝑥𝑛 𝑣𝑛 = 0
has at least one solution, namely
𝑥1 = 0, 𝑥2 = 0, … , 𝑥𝑛 = 0.
If this the only solution, then 𝑆 is called linearly independent set. If
there are other solutions, then 𝑆 is called a linearly dependent set.
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EXAMPLE: A LINEAR DEPENDENT SET
If 𝑣1 = 2, −1, 0, 3 , 𝑣2 = 1, 2, 5, −1 , and 𝑣3 = (7, −1, 5, 8) ,
then the set of vectors 𝑆 = {𝑣1 , 𝑣2 , 𝑣3 } is linearly dependent,
since 3𝑣1 + 𝑣2 − 𝑣3 = 0.
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EXAMPLE: A LINEAR INDEPENDENT SET
Consider the vectors 𝑣1 = (1, 0, 0), 𝑣2 = (0, 1, 0), and 𝑣3 = (0, 0, 1) in
𝑅3 . In terms of components the vector equation 𝑥1 𝑣1 + 𝑥2 𝑣2 + 𝑥3 𝑣3 = 0
becomes
𝑥1 (1, 0, 0) + 𝑥2 (0, 1, 0) + 𝑥3 (0, 0, 1) = (0, 0, 0)
or equivalently,
𝑥1 , 𝑥2 , 𝑥3 = 0, 0, 0 , So the set 𝑆 = {𝑣1 , 𝑣2 , 𝑣3 } is linearly independent.
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HOW TO CALCULATE IT
Step 1 Step 2 Step 3
Construct a Find REF If REF has
matrix by
no zero LI
using
row
vectors
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HOW TO CALCULATE IT
Step 1 Step 2
Construct a Step 3
Find REF
matrix by If REF has LD
using zero row
vectors
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LINEAR DEPENDENCE RELATION
To check a set is
LD or LI
If set is LD
Then LDR is
possible
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MATHEMATICAL PROBLEMS
Problem-1: Determine whether the vectors 2, −1, 0, 3 , 1, 2, 5, −1 and 7, −1, 5, 8
are linearly independent or dependent. If dependent then find a LDR among them and
verify it.
Solution:
Let 𝑣1 = 2, −1, 0, 3 , 𝑣2 = 1, 2, 5, −1 and 𝑣3 = 7, −1, 5, 8 .
2 −1 0 3 𝑣1
Now, 1 2 5 −1 𝑣2
7 −1 5 8 𝑣3
2 −1 0 3 𝑣1
𝑅2′ ⟶ 𝑅1 − 2𝑅2
~ 0 −5 −10 5 𝑣1 − 2𝑣2 ′
0 −5 −10 5 7𝑣1 − 2𝑣3 3 ⟶ 7𝑅1 − 2𝑅3
𝑅
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MATHEMATICAL PROBLEMS
2 −1 0 3 𝑣1
~ 0 −5 −10 5 𝑣1 − 2𝑣2 𝑅3′ ⟶ 𝑅3 − 𝑅2
0 0 0 0 6𝑣1 + 2𝑣2 − 2𝑣3
Since there is a zero row in the REF, the given vectors are linearly dependent.
Therefore, there exists a linear dependence relation among the vectors.
LDR:
6𝑣1 + 2𝑣2 − 2𝑣3 = 𝟎
⇒ 6 2, −1, 0, 3 + 2 1, 2, 5, −1 − 2 7, −1, 5, 8 = 0, 0, 0, 0
⇒ 12, −6, 0, 18 + 2, 4, 10, −2 − 14, −2, 10, 16 = 0, 0, 0, 0
⇒ 0, 0, 0, 0 = 0, 0, 0, 0
Verified 9
MATHEMATICAL PROBLEMS
Problem-2: Determine whether the vectors 2, 3, 1, −3 , 2, 3, 1, −2 and 4, 6, 2, −3
are linearly independent or dependent. If dependent then find a LDR among them and
verify it.
Solution:
Let 𝑣1 = 2, 3, 1, −3 , 𝑣2 = 2, 3, 1, −2 and 𝑣3 = 4, 6, 2, −3 .
2 3 1 −3 𝑣1
Now, 2 3 1 −2 𝑣2
4 6 2 −3 𝑣3
2 3 1 −3 𝑣1
𝑅2′ ⟶ 𝑅1 − 𝑅2
~ 0 0 0 −1 𝑣1 − 𝑣2 ′
0 0 0 −3 2𝑣1 − 𝑣3 3 ⟶ 2𝑅1 − 𝑅3
𝑅
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MATHEMATICAL PROBLEMS
2 3 1 −3 𝑣1
~ 0 0 0 −1 𝑣1 − 𝑣2 𝑅3′ ⟶ 3𝑅2 − 𝑅3
0 0 0 0 𝑣1 − 3𝑣2 + 𝑣3
Since there is a zero row in the REF, the given vectors are linearly dependent.
Therefore, there exists a linear dependence relation among the vectors.
LDR:
𝑣1 − 3𝑣2 + 𝑣3 = 𝟎
⇒ 2, 3, 1, −3 − 3 2, 3, 1, −2 + 4, 6, 2, −3 = 0, 0, 0, 0
⇒ 2, 3, 1, −3 − 6, 9 3, −6 + 4, 6, 2, −3 = 0, 0, 0, 0
⇒ 0, 0, 0, 0 = 0, 0, 0, 0
Verified 11
EXERCISE
• Assess the linear independence of the following vectors.
• If they are dependent, find a linear dependence relation and verify it.
i. 1, 2, −1, 0 , 1, 3, 1, 2 , 6, 1, 0, 1
ii. −2, 6, −5 , 0, 5, −6 , −6, −18, 15 , −4, 6, −2
iii. 0, 3, 1, −1 , 6, 0, 5, 1 , 4, −7, 1, 3
iv. 2, 1, 3, −1 , 2, 3, 1, 2 , 3, 2, 5, 6 , −2, −7, 3, −8
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ANY QUESTION???
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