Ques 1.

 Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample. 

i) The function f : R → R defined by f(x) = cos x is 1-1.
ii) The operation ∗ defined by x ∗ y = log(xy) is a binary operation on S, where S is the
set {x ∈ R|x > 0}.
iii) The set  is a subspace of 

.
iv) There is no 7×5 matrix of rank 6.
v) If  and V 0 are vector spaces and T : V → V 0 is a  linear transformation, then
whenever u1,,...,uk are linearly independent, Tu1, Tu2, ..., Tuk are also linearly
independent.
vi) If V is a vector space and T : V → V is a linear operator with det(T) = 0, then T is
not diagonalisable.
vii) The degree of the minimal polynomial of a 3×3 matrix is at most 2.
viii) For any 2×2 matrix 
.
ix) The only matrix which is both symmetric and skew-symmetric is the zero matrix.
x) There is no co-ordinate transformation that transforms the quadratic form  to the quadratic form xz+yz.