Let denote the golden ratio. Solve the equation
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 .
Let be similar to . Does hold?
No! Take then . The matrices are similar but not equal.
Let -Vect, , -linear. Prove that
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).
Let denote the Fibonacci sequence such that and . Prove that
It holds that
It follows from Cassini’s identity that
Setting back at we get
Since we get
Evaluate the integral