Final answer:
The vectors in scenario a) and b) are linearly dependent due to relationships that allow one to be written as the sum of others in the set. However, the vectors in scenario c) are linearly independent as each new vector can be considered an extension of the original linearly independent set by adding multiples of previous vectors.
Step-by-step explanation:
The question regarding the linear independence of vectors requires us to assess if a set of vectors can be written as a linear combination of each other. Vectors are linearly independent if no vector in the set can be written as a linear combination of the others. Let's analyze each scenario given by the student:
a) v₁ - v₂, v₂ - v₃, v₁ - v₃
The given vectors can be transformed into v₁ - v₂ + v₂ - v₃ = v₁ - v₃. These are linearly dependent because one can be obtained by summing the other two.
b) v₁ - v₂, 1/2(v₂ - v₃), 1/3(v₃ - v₄),..., 1/6 (v₆ - v₁)
This is a mixture of scaled vectors that could potentially be linearly independent or dependent, but given that we can write the last vector, 1/6(v₆ - v₁), as a scalar multiple of the sum of all the previous vectors, it implies that they are linearly dependent.
c) v₁, v₂, + 2v₁, v₃+3v₂, v₄+4v₃,..., vₙ+nvₙ₋₁
Since each newly formed vector is the sum of a unique vector not represented in the previous ones plus a multiple of one of the previous vectors, they form an extended set of the original linearly independent set v₁, v₂,...,vₙ which means they are also linearly independent.