(1 point) Let V-R\" and let H be the subset of V of all points on the plane-2-5y
ID: 3116713 • Letter: #
Question
(1 point) Let V-R" and let H be the subset of V of all points on the plane-2-5y-5z = 10 Is H a subspace of the vector space V? Is H nonempty? 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and syntax such as , 4,5,6> 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated ist and syntax such as 2, . 4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your arkKi to pai13.Explanation / Answer
1.H is non-empty as the point (-5,0,0) is in H.
2.Let (x1,y1,z1) and (x2,y2,z2)be 2 points on the given plane. Then -2x1-5y1-5z1= 10 and -2x2-5y2-5z2= 10. Further, -2(x1+x2)-5(y1+y2)-5(z1+z2)= (-2x1-5y1-5z1)+( -2x2-5y2-5z2)= 10+10 = 20. This implies that the point (x1+x2,y1+y2,z1+z2) is not on the given plane. Hence H is not closed under vector addition. The points whose sum is not on the plane are <-5,0,0>,<0,-1,-1>.
3. Let the point (x1,y1,z1) lie on the plane and let k be an arbitrary scalar. Then-2x1-5y1-5z1= 10 and -2kx1-5ky1-5kz1= k(-2x1-5y1-5z1= 10)= k*10 = 10k 10 unless k = 1. This implies that the point (kx1,ky1,kz1) is not on the given plane. Hence H is not closed under scalar multiplication. An example is 4,<-5,0,0>.
4. H is not a subspace of V. H is not closed under either vector addition or scalar multiplication. Hence H is not a vector space, and, therefore, not a subspace of V = R3.
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