linear algebra question 4 vector space 7 Let H be the set of

linear algebra question 4

vector space

7) Let H be the set of all points in the xy-plane having at least one nonzero coordinate: H-Xx, 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

Solution

A) In zero vector both x and y are zero.

Hence 0 vector is not in H.

------------------------------------------------

B) Take two vectors (2 0) and (0 -2) in H

Adding we have (2, -2) both non zero not in H.

Not closed under addition

---------------------------------------

C) If c is a scalar then for (x,y) in H

c(x,y) = (cx, cy) with any one being 0.

Thus closed under scalar multiplication.

H is not a vector space.