Each of the following statements is either True or False If

Each of the following statements is either True or False. If True, give a brief justification. If False, provide an example showing that is not always true. If v_1, ..., v_4 are in R^4 and v_3 = 0 then {v_1, v_2, v_3, v_4} is linearly dependent. If v_1, ..., c_4 are in R^4 and v_3 is not a linear combination of v_1, v_2, and v_4 then v_1, v_2, v_3, v_4 is linearly independent. If v_1, ..., v_4 are linearly independent vectors in R^4 then { v_1, v_2, v_3} are also linearly independent.

Solution

(a) FALSE: The vectors are linearly dependent because v₃ is not in the basis of R⁴.

(b) TRUE: If v₃ is not a linear combination of v₁, v₂, v₄ then v₁, v₂, v₃, v₄ are linearly independent. Otherwise, v₃ could have been written as linear combination with non-zero cefficient to any one of the vectors v₁, v₂, v₄.

(c) TRUE: Every subset of linearly independent set is linearly independent.