Which of the following sets of vectors in R³ are linearly independent:

1. [(1, 0, 0), (0, 1, 0), (1, 1, 0)]

2. [(1, 0, 0), (0, 1, 0), (0, 0, 1)]

3. [(0, 1, 0), (1, 0, 1), (1, 1, 0)]

4. [(0, 0, 1), (0, 1, 0), (0, 1, 1)]

Concept:

Linearly dependence of Vectors: A set containing the vectors u₁, u_2, ....u_r defined over a field F is said to be linearly dependent if scalars a₁, a₂, ....a_r ∈ R (not all zero) such that,

⇒ a₁ u₁ + a₂ u₂ + ......+ a_r u_r = 0

In brief linearly dependent is written as 'L.D'.

Linear Independence of Vector: A set containing the vectors u1, u2, ....ur defined over a field F is said to be lineraly independent if it is not linearly dependent, I,e. if every equation of the form a1 u1 + a2 u2 + ......+ ar ur = 0 ⇒ a_i = 0

Calculation:

Let u₁ = (1, 0, 0), u₂ = (0, 1, 0), u₃ = (0, 0, 1)

Also, a₁ u₁ + a₂ u₂ +.....+a_r u_r = 0

This gives (a₁ , a₂ , a₃) = (0, 0 ,0)

⇒ a₁ = 0, a₂ = 0, a₃ = 0

⇒ a₁ + a₂ + a₃ = 0 

This proves that the vectors u₁, u₂, u₃ are linearly independent. 

Similarly for (0, 1, 0), (1, 0, 1), (1, 1, 0)

Let u1 = (0, 1, 0), u2 = (1, 0, 1), u3 = (1, 1, 0)

⇒ (a₂ + a₃ , a₁ + a₃, a₂) = 0

⇒ a1 = 0, a2 = 0, a3 = 0

⇒ a1 + a2 + a3 = 0 

This proves that the vectors u1, u2, u3 are linearly independent.