Question

In the following list of subsets of R³ , select the ones that are subspaces of R³ (multiple answers).

(As usual, incorrect answers will earn you negative points)

		Entire R3
		{ (x+y, x-y, y ) | x,y are real numbers }
		{ (x,y,z) | x,y,z are all nonnegative real numbers }
		{ ( x, -x, 2x ) | x is a real number }
		{ (1,1,1) }
		{ (x,y,3) | x,y are real numbers }
		{ (0,0,0) }
		{ (x+1, y+1, z+1) | x,y,z are real numbers }
		{ (x,y,0) | x,y are real numbers }
		{ (x,y,z) | x + y + z = 1 }

Answer 1

1. Entire R³ is a subspace of R³, as every vector space is a subspace of itself.                             
2. V = { (x+y, x-y, y ) | x,y are real numbers } is a subspace of R³, as V? R³, is closed under vector addition and scalar multiplication and also contains the zero vector.              
3. V ={ (x,y,z) | x,y,z are all non-negative real numbers } is not a subspace of R³, as it is not closed under scalar multiplication. ( k(x,y,z) = (kx,ky,kz) ? V when k is negative).                     
4. V = { ( x, -x, 2x ) | x is a real number } is a subspace of R³, as V? R³, is closed under vector addition and scalar multiplication and also contains the zero vector.              
5. V = { (1,1,1) } is not a subspace of R³, as V? R³, is not closed under either vector addition or scalar multiplication and also does not contain the zero vector.                
6. V = { (x,y,3) | x,y are real numbers } is not a subspace of R³, as V? R³, is not closed under vector addition and also does not contain the zero vector.                           
7. V = { (0,0,0) } is a subspace of R³, as V? R³, is closed under vector addition and scalar multiplication and also contains the zero vector.                            
8. V = { (x+1, y+1, z+1) | x,y,z are real numbers } is not a subspace of R³, as V? R³, is not closed under either vector addition or scalar multiplication and also does not contain the zero vector.                        
9. V = { (x,y,0) | x,y are real numbers } is a subspace of R³, as V? R³, is closed under vector addition and scalar multiplication and also contains the zero vector.                              
10. { (x,y,z) | x + y + z = 1 } is not a subspace of R³, as V? R³, is not closed under either vector addition or scalar multiplication and also does not contain the zero vector.