.X 4. (5 points) Let H be the subset of vectors in R3 where H = y :x+y+z = 1 under the Z operations .761 I2 .761 +JC2 1 3'1 69 3'2 = y1 +y2 Z1 Z2 Z1 +Z2 x rx r + 1 r9 y = 0* Z 1'2 You can assume that H is closed under EB and G, and that Vector Space Properties (l)(2), (5)-(8) are true. This problem asks you to prove that properties (3) and (4) are true. a. (3 points) Show that there is an element 0 in H where 0 EB u = u = I] {B 0 for all u in H. b. (2 points) Show that for each element 11 in H , there exists an element u in H where 11 EB u = u 69 u = 0 A couple hints for Problem 4: 0 a 0 The zero vector 0 for H is NOT the usual zero vector 0 . Thus, it's unknown: 0 = b . Utilize 0 C that as a starting point for searching for 0. 0 You really can't do part (b) until you gure out part (a). Your answer for the zero vector will guide your answer for u. Again, u is NOT the usual additive inverse. 1that means examples with numbers, not variables 2If u or v are any vectors in W, uBv is in W 31f 6 is a real number and I1 is any vector in W, then 9011 is in W