Vector Space

Which of the following are generators of the multiplicative group {(1,2,3,4,5,6), x₇} where x₇ denotes multiplication moduls 7?

In a group G, if a⁵ = e, aba^-1 = b² for a, b ∈ G then o(b) is equal to:

Which one of the following statements is wrong.

Which one of the following is wrong?

If W is a subspace of R³, where W = {(a, b, c): a+b+c = 0}, then dim W is equal to

Which of the following is incorrect?

Given below are two statements
Statement I: Every cyclic group is abelian
Statement II: (Z,+) is a cyclic group with 1 and -1 as the only generators
In the light of the above statements, choose the most appropriate answer from the options given below

Given below are two statements
Statement I: Let G a finite group and H a subgroup of G. Then, the order of H is a divisor of the order of G. That is, |H| divides |G|
Statement II: Let a be an element in a finite group G. Then, O(a) divides |G|
In the light of the above statements, choose the most appropriate answer from the options given below

Let be the linear mapping defined by:
F(x,y,z,t)=(x-y+z+t, 2x-2y+3z+4t, 3x-3y+4z+5t), then nullity (F) equals

If and are the 3-dimensional subspaces of the 4-dimensional vector space, then the smallest possible dimension of is

A. Suppose U and W are finite-dimensional subspace of a vector space V, then
B. let ,W = {(a,b,c:a≥0)}, then W is a subspace of V
C. If u =(1,2),v=(3,-5), then u and v are linearly independent
D. (1,1,1) and (1,0,1) form a basis of
choose the most appropriate answer from the options given below

If and , then grad is equal to :

If is irrotational, where a, b and c are constant, then a² + b² + c² is equal to :

The directional derivative of Φ(x,y,z) = x²yz+4xz² at (1, -2, 1) in the direction of is

Let U and W are distinct 4-dimensional subspaces of a vector space V of dimension 6. Consider the following statements:
A. The dimension of U ∩ W is either 2 or 3.
B. The dimension of U + W is either 5 or 6.
C. The dimension of U ∩ W is always greater than 4.
D. The dimension of U + W is always greater than 4.
Choose the correct answer from the options given below: