Let A - {v_1,v_2} be a linearly independent set in R^3 and w be a vector in R^3.

Question

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

Explanation / Answer

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

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