Linear Dependence And Linear Independence

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, …, α_n ∈ V are said to be linearly dependent if there exists a set of scalars c₁, c₂, …, c_n ∈ F, not all zero (where zero is additive identity of F), such that, c₁ α₁ + c₂ α₂ + … + c_n α_n = θ

Linear Independence

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

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 .

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.

A collection of vectors containing null vector is linearly dependent.

Proof:

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

We can write 0.α₁ + 0.α₂ + … + 0.α_r + 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, …, α_r, θ are linearly dependent.

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

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.

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

The number of n-tuples (a₁₁, a₁₂, …, a_1n), (a₂₁, a₂₂, …, a_2n), …, (a_n1, a_n2, …, a_nn) will be independent if and only if the determinant

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

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