Consider the infinite system of homogeneous linear equations: X_0 + x_1 - x_2 =
ID: 1720148 • Letter: C
Question
Consider the infinite system of homogeneous linear equations: X_0 + x_1 - x_2 = 0 x_1 + x_2 - x_3 = 0 X_2 + X_3 - X_4 = 0 x_3 + x_4 - x_5 = 0 X_i + X_i+1 - X_i+2 = 0 Show that the solutions of this homogeneous system of linear equations form a subspace of the space of all infinitely-tall vectors. Show that there are unique solutions p and q to this system that begin with Show that the two solutions p and q you found in part (b) form a basis for the space of all solutions for this homogeneous system.Explanation / Answer
(a) Let a[0],a[1],a[2]......
and b[0],b[1],b[2],.......
be two solutions of the system of equations.
It is clear that c[i] = m a[i] + n b[i] is a solution of the system for scalars m and n:
For example
c[0] + c[1] -c[2] = m( a[0]+a[1]-a[2]) +n ( b[0]+b[1]-b[2]) =0
the same consideration applies to all other equations in the system.
Thus this set of solutions is a subspace of infinitely tall vectors.
(b) Notice that p and q are constructed satisfying the given system of equations with the initial conditions
p[0] =0 , p[1] =1( thus p[2] =1, p[3] = 2...and so on.Basically Fibonacci sequence)
q[0] =1 , q[1] =0 ( thus q[2] =1, q[3] = 2......)
(c) Notice that the first two terms x[0] and x[1] determine the subsequent terms of the solution completely.
So the dimension of this subspace is 2.
Now the solutions p[n] and q[n] are linearly independent .
Otherwise let Ap[n] + Bq[n] = 0 fro all n
For n = 0 and 1 this gives
A .1 + B.0 = 0
and
A.0 + B.1 =0
Thus A =0 and B =0.
Thus p and q form a basis for the set of all solutions of the given system