Let A v1v2 be a linearly independent set in R3 and w be a v

Let A - {v_1,v_2} be a linearly independent set in R^3 and w be a vector in R^3. Determine which of the following is true and provide the correct justification. (a) The set {v_1, v_2, w} is always linearly independent. Prove it (b) The set {v_1,v_2,w} is sometime linearly independent and sometimes dependent. Give an example of v_1,v_2, w_1 and w_2 with {v_1, v_2, w_1) linearly independent, while {v_1, v_2, w_2} is linearly dependent. (c) The set {v_1, v_2, w} is always linearly dependent. Prove it

Solution

The vector w cannot be defined as a linear combination of the other vectors, then the set v₁,v₂ and w is always linearly independent

The answer is A