Pythagorean Prove that if the functions f and g are orthogon

Pythagorean Prove that if the functions f and g are orthogonal over [a, b], then ||f + g||^2 = ||f||^2 + ||g||^2 over [a, b]. Confirm the group axioms for the special linear group. SL(n, IR)= {n times n matrices A| det A = 1}

Solution

||f+g||^2 = <f+g,f+g>(inner product)

= <f,g> + <g,f> + <f,f> + <g,g>

=||f||^2 + ||g||^2 (since f and g are orthogonal <f,g> is zero)

6) SL(n,R) is a group under multiplication

since if |A| = 1 and |B| = 1

|A.B| = |A|.|B| = 1 closure property

A.I = A , I is identity matrix ( Identity exists for the group and det(I) = 1)

if |A| = 1

A is invertible , there exists B such that

AB = I the identity matrix

B is unique and |B| = 1