Question

The statement is either true in all cases or false. If false, construct a specific example to show that the statement is not always true. If v_1, ......, v_4 are in R^4 and v_3 = 2v_1 + v_2, then {v_1, v_2, v_3, v_4} is linearly dependent. Choose the correct choice below and, if necessary, fill in the answer boxes to complete your choice. a. True Because v_3 = 2v_1 +v_2, v_4 must be the zero vector. Thus, the set of vectors is linearly dependent. b. True. The vector v_3 is a linear combination of v_1 and v_2, so at least one of the vectors in the set is a linear combination of the others and set is linearly dependent. c. True. If c_1 =2, c_2 = 1, c_3 = 1, and c_4 = 0, then c_1v_1 + middot middot middot + c_4v_4 =0. The set of vectors is linearly dependent. d. False. If v_1 =, v_2 =, v_3 =, and v_4 = [1 2 1 2], then v_3 = 2v_1 + v_2 and {v_1, v_2, v_3, v_4} is linearly independent.

Answer 1

Answer: The correct option is (B).

True. The vector V₃ is a linear combination of V₁ and V₂, so at least one of the vectors in the set is a linear combination of the others and set is linearly dependent.  

Step-by-step explanation:  

Given that;

If V₁ ....... V₄ are in R⁴ and V₃ = 2V₁ + V₂ then {V₁, V₂, V₃, V₄} is linearly dependent.  

Lets {V₁, V₂, V₃, V₄}  are linearly dependent

Then there exist a scalars C₁, C₂, C₃, C₄

So that C₁V₁ + C₂V₂ + C₃V₃ + C₄V₄ = 0

where at least one of the Ci ≠ 0.  

Take C₃ ≠  0 then we have V₃ = (C₁V₁ + C₂V₂ + C₄V₄) / C₃

V₃ is a linear combination of V₁, V₂ and V₄}

that is

Given V₃ = 2V₁ + V₂

⇒ 2V₁ + V₂ - V₃ = 0

⇒ 2V₁ + V₂ + V₃ + 0V₃ = 0

Here, C₁ = 1, C₂ = 1,  C₃ = -1 and C₄ = 0

So that {V₁, V₂, V₃, V₄} is linearly dependent.

therefore option B id the right answer.

- True. The vector V₃ is a linear combination of V₁ and V₂, so at least one of the vectors in the set is a linear combination of the others and set is linearly dependent.