जाँच कीजिए कि x∈ R\{2} के लिए (fog) (x) = x और x∈ R \{-1} के लिए (fog) (x) = x

For the following matrices, check whether there exists an invertible matrix P such that P^-1AP is diagonal. When such a P exists, find P.

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

i) The function  defined by f(x) = cosx 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 {(x₁.x₂,...,xn) X₁,X₂,...,xn ∈ R,X₁ = 2x2+3} is a subspace of Rⁿ

iv) There is no 7 x 5 matrix of rank 6.

v) If V and V' are vector spaces and T: V→ V' is a linear transformation, then whenever u₁,u₂,..., u_k are linearly independent, T_u₁, T_u2, ... Tu_k 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 x 3 matrix is at most 2.

viii) For any 2 x 2 matrix A, Adj (A') = (Adj(A))'.

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 x²+y² +z² to the quadratic form xz+yz.

Find the direction cosines of the perpendicular from the origin to the plane 

Check that g: R\{2} → R given by g(x) = is well defined and 1 - 1. Further, check that g(x) -1 for any x ∈ R.

जाँच कीजिए T(W)⊂ W

Let V be the set of all functions that are twice differentiable in R and

S = {cosx, sinx, xcosx,xsinx}.