Final answer:
The question relates to the concept of a spanning set in linear algebra, and the answer clarifies that the given sets cannot span ℝ² since they are not vectors but scalar values. Option B is correct.
Step-by-step explanation:
In mathematics, specifically linear algebra, a spanning set for a vector space, such as ℝ², is a set of vectors from which any vector in the space can be represented as a linear combination of the vectors in the set. For a set to be a spanning set of ℝ², it must contain at least two non-collinear vectors.
Both sets given in the question, (a) {1, 2} and (b) {-1, -2}, consist of scalar values, not vectors, and therefore cannot form a basis for ℝ². Consequently, the answer is (b) No, for both. In ℝ², we would typically expect to see vectors such as (1, 0) and (0, 1) or any pair of non-collinear vectors to serve as a spanning set.
To determine if a set is a spanning set for ℝ², we need to check if any vector in ℝ² can be written as a linear combination of the vectors in the set.
a) The set {1, 2} is not a spanning set for ℝ² because it only contains a single vector, and any vector in ℝ² cannot be written as a linear combination of that vector alone.
b) The set {-1, -2} is also not a spanning set for ℝ² because it also only contains a single vector.