Final answer:
The statement that 'n' vectors in ℝ^n are necessarily linearly dependent is false, as demonstrated by the linear independence of the canonical basis vectors in ℝ^n.
Step-by-step explanation:
The question seems to be asking for a determination of whether a given mathematical statement about vectors and linear dependence is true or false. Specifically, if vectors v1, v2, ..., vn are in ℝn and n ≥ 2, are they necessarily linearly dependent? To answer this, we will consider the definition of linear dependence.
A set of vectors in ℝn is considered linearly dependent if there are scalars, not all zero, such that a linear combination of these vectors equals the zero vector. In contrast, if the only way to obtain the zero vector from a linear combination of the vectors is by having all scalars be zero, the vectors are linearly independent.
Counterexample for Linear Dependence
One may think that if there are 'n' vectors in the space ℝn, they must be linearly dependent. However, this is false. For example, the canonical basis vectors of ℝn (e1 = [1,0,0,...,0], e2 = [0,1,0,...,0], ..., en = [0,0,...,1]) are in ℝn, and there are 'n' of them, yet they are linearly independent.
Thus, having 'n' vectors in ℝn is not a sufficient condition for those vectors to be linearly dependent. The proper analysis of the vectors is required to determine whether they are linearly dependent or independent.