(1 point) Let V = R3 and let H be the subset of V of all points on the plane 7x
ID: 3116717 • Letter: #
Question
(1 point) Let V = R3 and let H be the subset of V of all points on the plane 7x + 2y-4z = 14, Is H a subspace of the vector space V? 1. Is H nonempty? choose 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 , 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a vector irn H whose product is not in H, using a comma separated list and syntax such as 2, Is H a subspace of the vector space V7 You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. 4. chooseExplanation / Answer
1.H is non-empty as the point (2,0,0) is in H.
2.Let (x1,y1,z1) and (x2,y2,z2)be 2 points on the given plane. Then 7x1+2y1-4z1= 14 and 7x2+2y2-4z2= 14. Further, 7(x1+x2)+2(y1+y2)-4(z1+z2)= (7x1+2y1-4z1)+( 7x2+2y2-4z2)= 14+14 = 28. 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 <2,0,0>,<2,2,1>.
3. Let the point (x1,y1,z1) lie on the plane and let k be an arbitrary scalar. Then 7x1+2y1-4z1= 14 and 7kx1+2ky1-4kz1 = k(7x1+2y1-4z1)= k*14 = 14k 14 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,<2,0,0>.
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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