Help with part B only Show that the vectors v1 1 2 3 4 v2

Help with part B only

Show that the vectors v_1 = (1, 2, 3, 4), v_2 = (0, 1, 0, -1), and v_3 = (1, 3, 3, 3) form a linearly dependent set in I R^n. Express each vector in part i) as a linear combination of the other two. Prove that if{v_1, v_2, v_3} is a linearly independent set of vectors, then so are {v_1, v_3}, and {v_2}.

Solution

(b) It is given that {V₁,V₂,V₃} is a linearly independent set of vector then

we have x₁v₁ + x₃v₃ = 0

and x₂v₂ = 0

Since v₁, v₂, and v₃ are linearly independent, we must have the coefficients of the linear combination equal to 0, that is, we must have

x₁ + x₃ = 0

and x₂ = 0

from which it follows that we must have x₁ = x₂ = x₃ = 0.

Hence {v₁,v₃}and {v₂} are also linearly independent