Polynomial equation
Let denote the golden ratio. Solve the equation
Solution
First of all we note that
We easily note that is one root of the equation, hence using Horner we get that
Hence is a double root and the other root is
.
Similarity implies equality?
Let be similar to
. Does
hold?
Solution
No! Take then
. The matrices are similar but not equal.
On tensors
Let -Vect,
,
-linear. Prove that
Solution
Recall the general definition of the tensor product of linear maps, we have successively:
Thus, the two linear maps are equal when composed with the canonical bilinear map
, hence equal (by the universal property).
Fibonacci series
Let denote the Fibonacci sequence such that
and
. Prove that
Solution
It holds that
It follows from Cassini’s identity that
(1)
Setting back at
we get
(2)
Since we get
(3)
Hence,
Logarithmic integral
Evaluate the integral