Question

Determine whether the set S spans ℝ². If the set does not span ℝ², then give a geometric description of the subspace that it does span. S = {(1, 1), (−2, 2)}. S spans ℝ².

Answer 1

To determine whether the set S = {(1, 1), (-2, 2)} spans R^2, we check if every vector in R^2 can be written as a linear combination of the vectors in S. By solving the equations, we find that S spans R^2.

Step-by-step explanation:

To determine whether the set S = {(1, 1), (-2, 2)} spans ℝ², we need to check if every vector in ℝ² can be written as a linear combination of the vectors in S.

Let (x, y) be any vector in ℝ². We want to find scalars a and b such that a(1, 1) + b(-2, 2) = (x, y).

Simplifying this equation, we get (a - 2b, a + 2b) = (x, y). Comparing the components, we have: a - 2b = x and a + 2b = y.

Solving these two equations simultaneously, we find a = (x + y)/2 and b = (y - x)/4. Therefore, we can express any vector (x, y) in terms of the vectors in S, which means S spans ℝ².