LINEAR ALGEBRA (Math 211)

**HW #17** (Due Wed. Dec. 6 )

**1.** Find the dimensions and bases for all eigenspaces of
the following operators

The purpose of the following problems 2-6 is to derive a closed formula
for Fibonacci numbers *F*_{n}
(*F*_{0}=1, *F*_{1}=1,
*F*_{n}= *F*_{n-1}+*F*_{n-2} for all integers *n*>2).
The ``closed formula" means not a recurrent formula as above which
expresses
*F*_{n} in terms of previous numbers, but rather a formula which expresses
*F*_{n} in terms of *n* only.

**2.** Prove that

**3.** Suppose that the matrices *A* and *B* are similar. So there exist
an invertible matrix *P* such that
.
Prove that
for any integer *n*.

**4.** Diagonalize the matrix
from the problem 2. This means find an invertible matrix *P* and a
diagonal matrix *B* such that
.

**5.** (a) Find the matrix *B*^{n} for *B* from the previous problem.

(b) Using the problem 3 find *A*^{n} for *A* from the previous
problem.

**6.** Find the closed formula for Fibonacci number *F*_{n}.