Question

Let V = R3 be a vector space and let H be a subset of V defined as H = {(a, b,c) : a? = b2 = c}, then H Select one: O A. satisfies only first condition of subspace B. is a subspace of R3 C. None of them O D. satisfies second and third condition of subspace

Answer 1

$\text{ Three conditions of subspace are }$

A subset H of vector space V will be a subspace if it satisfies

(1) Additive identity, OH

(2) If r, y eH then x + y EH

(3) If I eH and a eR then ar eH

Given

$H =\left \{ (a,b,c) :a^2=b^2=c \right \}$

(1)

$\text{ Since, }0^2=0^2=0$

$=> (0,0,0)\epsilon H$

$\text{H satisfies first condition of subspace }$

$(2,2,4)\epsilon H, \:\:\:\: (3,3,9)\epsilon H$

$\text{ but }(2,2,4)+ (3,3,9) =(5,5,13)\notin H$

$\left ( \because 5^2=5^2 \neq 13 \right )$

$\text{H does not satisfy second condition of subspace }$

(3)

$(2,2,4)\epsilon H, \text{ Take }a =2$

$a(2,2,4) = 2(2,2,4)=(4,4,8)\notin H$

$\left ( \because 4^2=4^2 \neq 8 \right )$

$\text{H does not satisfy third condition of subspace }$

Hence,

$\text{H satisfies only the first condition of subspace }$

$=> \text{Option " A " is correct. }$