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={(0,2),(1,6)}

A.S spans R2
B.S does not span R2. S spans a line in R2.
C.S does not span R2. S spans a point in R2

Answer 1

The set S={(0,2),(1,6)} spans R2 because any vector in R2 can be expressed as a linear combination of (0,2) and (1,6). However, the vectors are not linearly independent.

Step-by-step explanation:

To determine whether the set S spans R2, we need to see if any vector in R2 can be written as a linear combination of the vectors in S. The set S is given as S = {(0,2), (1,6)}.

Let's suppose there is a vector (x, y) in R2. If S spans R2, then there must exist scalars a and b such that a(0,2) + b(1,6) = (x, y). This gives us the equations:

• 0 · a + 1 · b = x
• 2 · a + 6 · b = y

From the first equation, we see that b = x. Substituting b into the second equation, we get 2a + 6x = y. This system of equations has solutions for any real numbers x and y, so S does indeed span R2.

However, it's important to notice that the vectors in S are not linearly independent since (1,6) is a scalar multiple of (0,2). Therefore, even though they can span the space, they do not form a basis for R2.