Provide definition of orthogonality

Question

Question 1: If G is a group and H is a subgroup of G, then H is a normal subgroup of G if ghg^-1 ∈ H for all g from the set of generators of G and for all h from the set of generators of H.

Question 2: Let A be a skew symmetric n × n -matrix with entries in R i.e. A^T = -A then prove that

a) uTAu = 0 for every uRn.

b) In + A is an invertible matrix.

c) Give an example of a skew symmetric 2 × 2 matrix B with entries in C for which I2 + B is not invertible.

Question 3: Let two vectors x = (x₁, x₂, . . . , x_n) and y = (y₁, y₂, . . . , y_n).

a) Provide definition of Orthogonality.

b) Prove that if x and y are mutually orthogonal, then they are linearly independent.

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