Question

Determine if the statement is true or false, and justify your answer.

If u₄ is not a linear combination of {u₁, u₂, u₃}, then {u₁, u₂, u₃, u₄} is linearly independent.

A) False. Consider u₁ = (1, 0, 0), u₂ = (0, 1, 0), u₃ = (0, 0, 1), u₄ = (0, 1, 0).

B) False. Consider u₁ = (1, 0, 0), u₂ = (1, 0, 0), u₃ = (1, 0, 0), u₄ = (0, 1, 0).

C) True. The echelon form of the augmented matrix [u₁u₂u₃u₄] will have at least one row of zeroes at the bottom, which means the vectors are linearly independent.

D) False. The echelon form of the augmented matrix [u₁u₂u₃u₄] will have at least one row of zeroes at the bottom, which means the vectors are linearly dependent.

E) True. If {u₁, u₂, u₃, u₄} is linearly dependent, then u₄ = x₁u₁ + x₂u₂ + x₃u₃, which is a contradiction.

Answer 1