Question

determine whether the set S spans R³ . If the set does not span R³ , give a geometric description of the subspace that it does span. S={(4,7,3),(−1,2,6),(2,−3,5)}

Answer 1

To determine whether the set S spans R³, we need to check if every vector in R³ can be written as a linear combination of the vectors in set S. In this case, S does not span R³.

Step-by-step explanation:

To determine whether the set S spans R³, we need to check if every vector in R³ can be written as a linear combination of the vectors in set S.

1. Let's take a random vector in R³, say (x, y, z).
2. We need to find the scalars a, b, and c such that a(4, 7, 3) + b(-1, 2, 6) + c(2, -3, 5) equals (x, y, z).
3. Solving the system of equations, we get x = 2a - b + 2c, y = 7a + 2b - 3c, and z = 3a + 6b + 5c.
4. If there exist values of a, b, and c that satisfy these equations for every (x, y, z) in R³, then the set S spans R³. Otherwise, it does not span R³.

Therefore, to determine if S spans R³, we need to check if the system of equations has a solution for any (x, y, z) in R³. If there is no solution, that means the set S does not span R³.

In this case, S does not span R³.

Geometrically, the set S spans a subspace in R³, which can be visualized as a plane passing through the origin since it is a two-dimensional subspace. The vectors in S lie on this plane.