Linear Dependence and Linear Independence of Vectors


Linear Algebra / Sunday, October 28th, 2018
(Last Updated On: November 8, 2018)

In this article we will learn linear dependence and linear independence of vectors.

Linear Dependence

For a vector space V defined over a field F, the n vectors α1, α2, …, αn ∈ V are said to be linearly dependent if there exists a set of scalars c1, c2, …, cn ∈ F, not all zero (where zero is additive identity of F), such that, c1 α1 + c2 α2 + … + cn αn = θ

Linear Independence

For a vector space V defined over a field F, the n vectors α1, α2, …, αn ∈ V are said to be linearly independent if and only if c1 α1 + c2 α2 + … + cn αn = θ, ci ∈ F (i=1, 2, …, n) implies that c1 = c2 = … = cn = 0

 Example 01

The coordinate vectors α1= (1, 1, 0), α 2= (3, 2, 1) and α 3 =(2, 1, 1) are linearly dependent if there exists a set of scalars c1, c2, c3 not all zero, such that c1 (1, 1, 0) + c2 (3, 2, 1) + c3 (2, 1, 1) = (0, 0, 0).

This requires that

c1 + 3c2 +2c3 = 0

c1 + 2c2 +c3 = 0

c2 + c3 = 0

The system of homogeneous linear equations has non zero solution as the rank of the coefficient matrix is 2 (<3). We may also directly solve to check that c1 = 1, c2 = -1, c3 = 1 is a solution to the system. Hence, (1) α1 + (-1) α2 + (1) α3 = θ

Thus vectors α1 , α2, α3 are linearly dependent and any one of the vectors can be written as a linear combination of the other two. For example, α1 = α2 – α3 .

 Example 02

The coordinate vectors α1= (3, 2, 1), α 2= (0, 1, 2) and α 3 =(1, 0, 2) are linearly independent.

Suppose that c1, c2, c3 are scalars such that c1 (3, 2, 1) + c2 (0, 1, 2) + c3 (1, 0, 2) = (0, 0, 0).

This requires that

3c1 + c3 = 0

2c1 + c2 = 0

c1 + 2c2 + 2c3 = 0

It may be checked that the rank of the coefficient matrix is 3 = number of variables. Hence the only solution for the system of equations is c1 = c2 = c3 = 0. We can also directly obtain this solution.

Hence, by definition, it follows that the given vectors are linearly independent.

 Theorem 01

A collection of vectors containing null vector is linearly dependent.

Proof:

Let α1, α2, …, αr, θ be the collection of vectors.

We can write 0.α1 + 0.α2 + … + 0.αr + 1.θ = θ + θ+ …. θ = θ

We see among the numbers 0, 0, …0, 1 at least one, namely 1 is non-zero.

So, α1, α2, …, αr, θ are linearly dependent.

 Theorem 02

A collection of vectors which contains a collection of linearly dependent vectors is linearly dependent.

 Illustration

The vectors (1, 2, 3), (2, 4, 6), (5, 9, 1), (-6, 7, 8) and (11, 2, 5) are linearly dependent.

We see 2. (1, 2, 3) + (-1). (2, 4, 6) = (0, 0, 0)

So, (1, 2, 3), (2, 4, 6) are linearly dependent. So by above theorem the given five vectors are also linearly dependent.

 Theorem 03

Any part of a collection of linearly independent vectors is linearly independent.

 Theorem 04

The number of n-tuples (a11, a12, …, a1n), (a21, a22, …, a2n), …, (an1, an2, …, ann) will be independent if and only if the determinant.

linear independence 01

 Illustration

The three vectors α = (1, 2, 1), β = (2, 3, 1) and γ = (2, 2, 0) are linearly dependent because the determinant

linear independence 02

 

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