Linear algebra Let H be the set of all points in the xyplane

Linear algebra

Let H be the set of all points in the xy-plane having at least one nonzero coordinate: H = {[^x_y ]: x, y not both zero} .Determine whether H is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy:

A: Contains zero vector

B: Closed under vector addition

C: Closed under multiplication by scalars

A) H is not a vector space; does not contain zero vector. Give a counter example or explain why?

B) H is not a vector space; fails to satisfy all three properties. Give 3 counter examples .

C) H is not a vector space; not closed under vector addition. Give a counter example .

D) H is not a vector space; does not contain zero vector and not closed under multiplication by scalars. Give 2 counter examples or explain why?

Solution

A. Does not contain 0 vector

B. Not closed under addition

Example. Two vectors in H are: [0,1],[0,-1]

Adding gives [0,0] which is not in H

C. Not closed under scalar multiplication

[1,0] is in H

0*[1,0] is not

B)