Find a linearly independent set of vectors

Question

Find a linearly independent set of vectors that spans the same subspace of R^3 as that spanned by the vectors

v₁ : < 2, -2, -3>

v₂ : <5, -4, -4>

v₃ : <-1 , 0 , -2>

Linearly independent set?

Answer 1

A spanning set can be reduced to a linearly independent set by removing any vectors from the set that are linear combinations of the other vectors in that set. {v₁,v₂} is a linearly independent set because clearly neither v₁ or v₂ is a scalar multiple of the other. However {v₁,v₂,v₃} is not linearly independent because v₃ is a linear combination of v₁ and v₂. Specifically 2v₁ - v₂ = v₃. So the set {v₁,v₂} suffices.