The world’s first farmers were not human beings but the leaf-cutter ants of the genus Atta. These ants do not eat the leaves they carry off. Instead, they bring them down into their nests, where they process the leaves into a pulp and then spread the pulp onto a fungus that grows only in their nests. The fungus then feeds on the food that has been brought to it and grows, after which the ants harvest the knob-like stalks of the fungus and consume them. What is the mode of interaction between the ants and the fungus in this community?

Show that the set F = ({1, 0}, +, .) is a field where + and . are defined as 1+1=0, 0+0=0,
0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.

Let X be a non-empty set and F be any field. Let XF be the set of all functions from X to
F. Show that XF is a vector space over F under the operations: (f + g)(x) = f(x) + g(x) and
(αf)(x) = αf(x).

Which of the following describes a limitation of the exponential growth model ?

A. Biomass growth is assumed to be not linked to the availability of nutrients or substrates

B. It does not predict when cell growth will cease

C. It predicts that all types of cells under all conditions will grow exponentially

D. All of the above are correct

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