Problem 1E

Exercises display sets in ℝ2. Assume the sets include the bounding lines. In each case, give a specific reason why the set H is not a subspace of ℝ2. (For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.)

Solution:-

Step1

Given that

Step2

To find

Assume the sets include the bounding lines. In each case, give a specific reason why the set H is not a subspace of ℝ2. (For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.)

Step3

We have

Diagram

Step4

As we can see that v=(1,1) is in given set, but -1.v=(-1,-1) is not in this set.