Chapter 3 Mathematical prerequisites

3.1 Notations

3.1.1 Sets

  • \(\mathbb{N}\): set of natural numbers
  • \(\mathbb{R}\): set of real number
  • \(\mathbb{C}\): set of imaginary number
  • \(\mathbb{K}\): either \(\mathbb{R}\) or \(\mathbb{C}\)

3.1.2 Matrices

  • \(0_n\) null matrix, dimension \(n\)

  • \(I_n\) identity matrix, dimension \(n\)

  • eigen value
  • eigen vectors
  • trace

  • determinant

  • close or open circle

  • spiral outward / inward / center

seqx <- seq(0, 1, .01)
plot(seqx, se)

3.2 Equations

3.2.1 Order 2

3.2.1.1 General equation

\[ ax^2 + bx + c = 0 \]

where \(a, b, c\) are real numbers.

3.2.1.2 Solution

\[(S) \iff x^2 + \frac{b}{a}x + \frac{c}{a} = 0\]

\[(S) \iff \left( x + \frac{b}{2a} \right)^2 - \left(\frac{b}{a}\right)^2 + \frac{c}{a} = 0\]

\[(S) \iff \left( x + \frac{b}{2a} \right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} = 0\]

\[(S) \iff \left( x + \frac{b}{2a} \right)^2 - \frac{b^2 + 4ac}{4a^2} = 0\]

\[(S) \iff \left( x + \frac{b}{2a} \right) = \pm \frac{\sqrt{b^2 + 4ac}}{2a} = 0\]

So:

\[x = \frac{-b \pm \sqrt{b^2 + 4ac}}{2a} \]

Note that \(b^2 + 4ac\) is called the discriminant.

3.2.1.3 Example: the golden ratio

Never heard about it? Have a look at the wikipedia page

So basically the golden ration verifies:

\[ \frac{b}{a+b} = \frac{a}{b} \]

which is equivalent to:

\[ b^2 - ab - a^2 = 0 \]

Let’s assume \(a = 1\) and we are looking for \(b\), solutions are then:

\[ b^2 - b - 1 = 0 \]

\[ b = \frac{1 \pm \sqrt{5}}{2} \]

The positive solution is the golden ratio!

3.3 Change of the basis

3.4 Eigen value

3.4.1 Trick in 2 D

lamda1 + lamda2 = trace lamba1 lambda2 = determinate

3.4.2 Ressources