Subspace of a Vector Space
Consider the vector space R2 over R. Then which of the following are subspaces of R2?
i. y=x
ii. xy=0
iii. x2+y2=9
iv. y=x2
i. y=x
ii. xy=0
iii. x2+y2=9
iv. y=x2
Consider the vector space R2 over R. Then which of the following are subspaces of R2?
i. {(x, y) ϵ R2 : x2 + y2 = 0}
ii. {(x, y) ϵ R2 : x.y>0}
iii. {(x, y) ϵ R2 : x>0, y<0}
i. {(x, y) ϵ R2 : x2 + y2 = 0}
ii. {(x, y) ϵ R2 : x.y>0}
iii. {(x, y) ϵ R2 : x>0, y<0}
Consider the vector space R3 over R. Then which of the following are subspaces of R3?
i. A sphere centered at origin with radius 1
ii. A plane passing through origin
iii. A line passing through origin
iv. A plane parallel to the x-y plane and passing through (1, 1, 1)
i. A sphere centered at origin with radius 1
ii. A plane passing through origin
iii. A line passing through origin
iv. A plane parallel to the x-y plane and passing through (1, 1, 1)
Which of the following are subspaces of the vector space R3 over R?
i. W1={(2x, 3y+z, 4) : x, y, z ϵ R}
ii. W2={(2x, 1, 2) : x ϵ R}
iii. W3={(y, z, x) : x, y, z ϵ R}
i. W1={(2x, 3y+z, 4) : x, y, z ϵ R}
ii. W2={(2x, 1, 2) : x ϵ R}
iii. W3={(y, z, x) : x, y, z ϵ R}
Consider the vector space M2 x 2 over R. Which of the following is a subspace of M2 x 2?
i. W1={A ϵ M2x2 | det(A) = 0}
ii. W2={A ϵ M2x2 | a12 = a21}
i. W1={A ϵ M2x2 | det(A) = 0}
ii. W2={A ϵ M2x2 | a12 = a21}
(A) Arbitrary union of subspaces of a vector space is a subspace.
(B) Arbitrary intersection of subspaces of a vector space is a subspace.
(B) Arbitrary intersection of subspaces of a vector space is a subspace.