Vector Spaces

Walkthrough

Determine whether the vectors v₁ = (1, 0, 1), v₂ = (0, 1, 1), v₃ = (1, 1, 2) are linearly independent.

1. 1.Recall the definition: vectors are linearly independent if the only solution to c₁v₁ + c₂v₂ + c₃v₃ = 0 is c₁ = c₂ = c₃ = 0.
2. 2.Set up the equation: c₁(1,0,1) + c₂(0,1,1) + c₃(1,1,2) = (0,0,0).
3. 3.This gives the system: c₁ + c₃ = 0, c₂ + c₃ = 0, c₁ + c₂ + 2c₃ = 0.
4. 4.From equations 1 and 2: c₁ = −c₃ and c₂ = −c₃. Substituting into equation 3: −c₃ − c₃ + 2c₃ = 0, which is 0 = 0 — always true.
5. 5.Since c₃ is a free variable (not forced to be 0), there are non-trivial solutions. The vectors are linearly DEPENDENT (specifically, v₃ = v₁ + v₂).