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 = (-5, 2)

Answer 1

To determine if the set s = (-5, 2) spans R2, we check if any vector in R2 can be expressed as a linear combination of s. By equating the components, solving the resulting equations, and expressing x and y in terms of a and b, we show that s spans R2.

Step-by-step explanation:

To determine whether the set s = (−5, 2) spans R2, we can check if any vector in R2 can be expressed as a linear combination of s.

1. Let's take an arbitrary vector in R2, v = (a, b).
2. To express v as a linear combination of s, we need to find values of x and y such that x⋅(-5, 2) + y⋅(−5, 2) = (a, b).
3. By equating the corresponding components, we get -5x + -5y = a and 2x + 2y = b.
4. Simplifying these equations, we have -5x - 5y = a and 2x + 2y = b.
5. If we divide both sides of each equation by -5 and 2 respectively and simplify, we obtain x = -a/5 -y and y = b/2 - x.
6. Since the values of x and y can be expressed in terms of a and b, the set s spans R2.

Therefore, the set s = (-5, 2) spans R2.