In the following list of subsets of R 3 , select the ones that are subspaces of
ID: 3137529 • Letter: I
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 }
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 }
Explanation / Answer
Entire R3 is a subspace of R3, as every vector space is a subspace of itself. 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. 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). 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. 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. 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. 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. 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. 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. { (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.
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