Final Answer:
No, the set {∈, ∈} is linearly dependent. For a set of vectors to be linearly dependent, there must exist scalars (not all zero) such that a linear combination of the vectors equals zero.
Step-by-step explanation:
For a set of vectors to be linearly dependent, there must exist scalars (not all zero) such that a linear combination of the vectors equals zero. Let's consider the set {∈, ∈}. If we take the scalars a and b such that a∈ + b∈ = 0, it implies that both a and b must be zero for this equation to hold true. However, this is impossible since at least one of the scalars must be non-zero for a linear combination to equal zero. Therefore, the only solution to a∈ + b∈ = 0 is a = b = 0, confirming that the set {∈, ∈} is linearly dependent.
In this case, the vectors in the set are essentially the same (both representing the same element). Any linear combination of them, such as a∈ + b∈ = 0, would imply that both scalars a and b must be zero to satisfy the equation, leading to a trivial solution. Consequently, there is no non-trivial combination that could result in zero without having all scalars equal to zero. Therefore, the set {∈, ∈} is linearly dependent due to the vectors being multiples of each other, making the set redundant in terms of independent information.
The linear dependence arises because the vectors in the set are not distinct, contributing redundant information. In linear algebra, a set of vectors is considered linearly dependent when one vector in the set can be expressed as a scalar multiple of another vector or a combination of other vectors in the set. In this scenario, both vectors represent the same element, hence any combination results in a trivial solution, confirming their linear dependence.