Determine whether the set S spans R3. If the set does not span R3, then give a geometric description of the subspace that it does span.S = {(1, 0, 3), (2, 0, −1), (4, 0, 5), (2, 0, 6)}S spans R3.S does not span R3. S spans a plane in R3.    S does not span R3. S spans a line in R3.S does not span R3. S spans a point in R3.

To determine whether the set S spans R3, we need to check if the vectors in S are linearly independent and if they can form any vector in R3.

The set S = {(1, 0, 3), (2, 0, -1), (4, 0, 5), (2, 0, 6)} contains four vectors in R3.

First, we can see that all vectors have the second component as 0. This means that they all lie in the same plane that is parallel to the x-z plane. Therefore, they cannot span R3, which requires the ability to reach any point in three-dimensional space, not just points in a single plane.

Next, we need to check if these vectors are linearly independent to determine the subspace they span. We can do this by setting up a system of linear equations to see if the only solution is the trivial solution.

However, we can also see that the vectors are not linearly independent by observation. For example, the third vector (4, 0, 5) can be obtained by adding the first vector (1, 0, 3) and the second vector (2, 0, -1). Similarly, the fourth vector (2, 0, 6) can be obtained by subtracting the second vector from the third.

Since the vectors are not linearly independent, they do not span the entire plane, but rather a line within the plane.

So, the set S does not span R3. Instead, it spans a line in the plane parallel to the x-z plane in R3.

To determine whether the set S spans R3, we need to check if the vectors in S are linearly independent and if they can form any vector in R3.

The set S = {(1, 0, 3), (2, 0, -1), (4, 0, 5), (2, 0, 6)} contains four vectors in R3.

First, we can see that all vectors have the second component as 0. This means that they all lie in the same plane that is parallel to the x-z plane. Therefore, they cannot span R3, which requires the ability to reach any point in three-dimensional space, not just points in a single plane.

Next, we need to check if these vectors are linearly independent to determine the subspace they span. We can do this by setting up a system of linear equations to see if the only solution is the trivial solution.

However, we can also see that the vectors are not linearly independent by observation. For example, the third vector (4, 0, 5) can be obtained by adding the first vector (1, 0, 3) and the second vector (2, 0, -1). Similarly, the fourth vector (2, 0, 6) can be obtained by subtracting the second vector from the third.

Since the vectors are not linearly independent, they do not span the entire plane, but rather a line within the plane.

So, the set S does not span R3. Instead, it spans a line in the plane parallel to the x-z plane in R3.