M208 TMA03 Question 4

Done in draft. Question about finding an orthogonal base of some abstract vector space. Not really difficult considering the marks assigned ( 20 ).

In MST121 / MS221 we learned to find the closed form for the Fibonacci sequence. Or solving the recurrence equation $a_{n} - a_{n-1} - a_{n-2} = 0$, equations of this type are called homogenous linear recurrence equations ( of the second order ). Inhomogenous equations have a function of n on the RHS. An inhomogenous recurrence equation of the first order is for example $a_{n} = 2 a_{n-1} + n$. The solutions of this equation consist of a homonegenous part and a particular part. ( More later. )