Which of the following are subspaces of R3 ?(i) {(x,y,z)| z = 2x+3y+2}(ii) {(x,y,z)| x2+y2=z2}Select one:a. (ii) onlyb. (i) and (ii)c. None of the other choices is correctd. (i) only

A subspace of R3 must satisfy three conditions:

1. The zero vector is in the subspace.
2. The subspace is closed under vector addition.
3. The subspace is closed under scalar multiplication.

Let's check these conditions for both options:

(i) {(x,y,z)| z = 2x+3y+2}

1. The zero vector (0,0,0) is not in this set because if x=y=0, then z=2 which is not zero.
2. & 3. Even if we check these, the first condition is not met, so this is not a subspace.

(ii) {(x,y,z)| x^2 + y^2 = z^2}

1. The zero vector (0,0,0) is in this set because if x=y=z=0, the equation holds.
2. The set is not closed under addition. For example, (1,0,1) and (0,1,1) are in the set, but their sum (1,1,2) is not.
3. The set is not closed under scalar multiplication. For example, (1,0,1) is in the set, but its double (2,0,2) is not.

So, neither (i) nor (ii) is a subspace of R3. Therefore, the correct answer is c. None of the other choices is correct.