Virtual Labs

Consider the vector space

R

² over

R

. Then which of the following are subspaces of

R

²?
i.

y=x

ii.

x y

=0
iii.

x

²+

y

²=9
iv.

y=x

a: Only (i) & (iii) Explanation

Explanation

b: Only (i) Explanation

Explanation

c: Only (ii), (iii) & (iv) Explanation

Explanation

d: (i), (ii), (iii) & (iv) Explanation

Explanation

Consider the vector space

R

² over

R

. Then which of the following are subspaces of

R

²?
i. {

(x, y)

R

² :

x

² +

y

² = 0}
ii. {

(x, y)

R

² :

x.y

>0}
iii. {

(x, y)

R

² :

x

>0,

y

<0}

a: Only (i) Explanation

Explanation

b: Only (i) & (iii) Explanation

Explanation

c: (i), (ii) & (iii) Explanation

Explanation

d: Only (iii) Explanation

Explanation

Consider the vector space

R

³ over

R

. Then which of the following are subspaces of

R

³?
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)

a: Only (i), (ii) & (iv) Explanation

Explanation

b: Only (ii) & (iii) Explanation

Explanation

c: Only (ii), (iii) & (iv) Explanation

Explanation

d: (i), (ii), (iii) & (iv) Explanation

Explanation

Which of the following are subspaces of the vector space

R

³ over

R

?
i.

W1

(2x, 3y+z, 4) : x, y, z

R

}
ii.

W

2={

(2x, 1, 2) : x

R

}
iii.

W3

(y, z, x) : x, y, z

R

}

a: Only (i) & (ii) Explanation

Explanation

b: Only (ii) & (iii) Explanation

Explanation

c: (i), (ii) & (iii) Explanation

Explanation

d: Only (ii) Explanation

Explanation

Consider the vector space

M

_{2 x 2} over

R

. Which of the following is a subspace of

M

_{2 x 2}?
i.

W1

A

M

_2x2 |

det(A) = 0

}
ii.

W2

A

M

_2x2 |

a

₁₂ =

a

₂₁}

a: Only (i) Explanation

Explanation

b: Only (ii) Explanation

Explanation

c: Both (i) & (ii) Explanation

Explanation

d: Neither (i) nor (ii) Explanation

Explanation

(A) Arbitrary union of subspaces of a vector space is a subspace.
(B) Arbitrary intersection of subspaces of a vector space is a subspace.

a: A is true, B is false Explanation

Explanation

b: A is false, B is false Explanation

Explanation

c: A is false, B is true Explanation

Explanation

d: A is true, B is true Explanation

Explanation

Subspace of a Vector Space