Answer to Question #52684 in Linear Algebra for ravish

Question

Which of the following statements are true and which are false? Justify your answer with a short proof or a counter example. i) ii) iii) iv) v) vi) vii) The function f:R→R define dbyf(x)=cosxis1-1. The operation∗defined by x∗y=log(xy)is a binary operation on S,where S is the set{x∈R|x>0}. The set{(x1.x2,...,xn)|x1,x2,...,xn∈R,x1=2x2+3}is a subspace of Rn. There is no 7×5 matrix of rank 6. If V and V are vector spaces and T:V→Visali near transformation,then whenever u1,u2,...,uk are linearly independent, Tu1,Tu2,...,Tuk are also linearly independent. If Visa vector space and T:V→Visali near operator with det(T)=0,then T is not diagonalisable. The degree of the minimal polynomial of a 3×3matrix is at most 2. viii) Forany 2×2 matrix A,Adj(At)=(Adj(A))t. ix) x) The only matrix which is both symmetric and skew-symmetric is the zero matrix. There is no co-ordinate transformation that transforms the quadratic form x2+y2+z2 to the quadratic form xz+yz.