Let X be any set and P(X) be the power set of X. Prove that P(X) is a vector space over
F = {0, 1} under the operation: A + B = A4B (symmetric difference) and αA = A if α = 1 and
is equal to φ if α = 0.

 

Find which of the axioms will be violated if addition of vectors in problem 3 is changed to A+B =
A
S
B 

In the following, find out whether S froms a subspace of V ?
(a) V = R3, S = {(x1, x2, x3) : x1 + 5x2 + 3x3 = 0}
(b) V = R3, S = {(x1, x2, x3) : x1 + 5x2 + 3x3 = 1}
(c) V = R3, S = {(x1, x2) : x1 ≥ 0, x2 ≥ 0}
(d) V = P(R), the set of all polynomials over reals and S = {p(x) ∈ P(R) : P(5) = 0}
(e) V = Rn, S = {(x1, x2, ..., xn) : x1 = x2}
 

Prove or disprove:
(a) Union of two subspaces of V is a subspace of V .
(b) Intersection of any number of subspaces is a subspace.

Check whether the following set of vectors are linearly dependent or independent.
(a) S = {(1, 2, −2, −1),(2, 1, −1, 4),(−3, 0, 3, −2)}
(b) S = {(1, 3, −2, 5, 4),(1, 4, 1, 3, 5),(1, 4, 2, 4, 3),(2, 7, −3, 6, 13)}

 Whether the system below is consistent? Justify.
x + 2y − 3z = 1
3x − y + 2z = 5
5x + 3y − 4z = 2

Solve
x + 2y − 3z + 2w = 2
2x + 5y − 8z + 6w = 5
3x + 4y − 5z + 2w = 4

For the system
x + 2y − z = 0
2x + 5y + 2z = 0
x + 4y + 7z = 0
x + 3y + 3z = 0 

Find the solution space as well as dimension.

The square root of 0.0081 is ………

What is the sum of 5x3 – 3x2 - 1 and 3x2 + 1?

All natural numbers and 0 are called the ……………….. numbers?