4. Algebra - JulTob/Mathematics GitHub Wiki

βš–οΈ Algebra

πŸ“ The Language of Abstraction and the study of abstract structures. How mathematical models are organized. From the Arabic $\color{#18f}Alβ€’Jabr$ ("reallocation")

πŸͺ¬ Algebra uses $\color{#189}symbols$ to represent entities, reducing our focus into one specific aspect we care about.

πŸ’« Equations & Expressions

  • πŸ’« The art of balance.

✨ Polynomials & Factorization

  • The secret structure of numbers.

β˜„οΈ Systems of Equations

  • How multiple truths tie together.

Algebra

Algebra is the study of mathematical procedures that combine symbols in order to express concepts.

Symbols

  • Values
    • Quantities
      • Numbers: οΌ‘, β„―, 𝛑, πš’
      • Constants: π‘Ž, 𝑏, 𝑐, n
      • Variables: π‘₯, 𝑦, 𝑧, πš›, 𝑑, 𝑓, πœ—
  • Operations
    • Addition οΌ‹, βˆ‘
    • Subtraction -
    • Multiplication Γ—, β‹…, (π‘Ž)(𝑏), π‘Ž(𝑏), π‘Žπ‘
    • Division Γ·, ∢, π‘Žβˆ•π‘, π‘βˆ£π‘Ž
    • Exponents π‘ŽΒ²,π‘Žα΅‡, ^, **
    • Roots Βͺ√, √, ³√
  • Relationship
    • Equality and inequality =, β‰ˆ, <, ≀, >, β‰₯, β‰ 
    • Proportionality ∝, ∷
    • Order β‰Ί, ≻, β‰Ύ, β‰Ώ, βŠ€, ⊁
  • Organization and order ( ), [ ], ⟨ ⟩, β§™β§˜, β§›β§š, Diagrams βŸ₯⟒⟜⫐

Expressions

Sets of symbols that represent a concept.

bh/2 2π›‘πš› sinπœƒ

Equation & Inequality

Relationships between Expressions.

β„―^2π›‘πš’ - 1 = 0 -1 ≀ cosπœƒ ≀ 1 𝑓(x) ∝ β„―Λ£

Linear Equation

π‘Žπ‘₯οΌ‹π‘οΌπŸΆ
    
π‘₯ = -𝑏/π‘Ž        

Quadratic Ecuations

π‘Žπ‘₯²+𝑏π‘₯οΌ‹π‘οΌπŸΆ

     -π‘Β±βˆšπ‘Β²-4π‘Žπ‘]
π‘₯ = ────────────
        2π‘Ž

Polynomials

Generalizing, the simplest type of functions are Polynomials.

𝑓(π‘₯)= π‘Žβ‚™π‘₯ⁿ .+. π‘Žπ‘₯²+𝑏π‘₯+𝑐
= βˆ‘π‘Žβ‚™Β·π‘₯ⁿ
= 𝚨 ∏(π‘₯-𝛼ₙ)

𝛼ₙ : Roots

A Polynomial of degree n has n solutions.

Fundamental theorem of Algebra.

Algebra Properties

Commutation

Commutative Properties: Order of the elements.

Association

Associative: Grouping of the elements.

Distributive

Distribution: Share to the elements

Order Of Operations

  1. Parentheses or brackets
  2. Exponents, Roots, absolute (left to right)
  3. Multiplication and division (left to right)
  4. Addition and subtraction (Left to right)

Sistemas Lineales

Sub Espacios vectoriales

Un espacio vectorial es una dimensiΓ³n formada por uno o mΓ‘s vectores (1, recta; 2, superficie; 3 volumen; 4 espacio-tiempo,5 niputaidea), o por una o mΓ‘s limitaciones (que son ecuaciones en las que se marca la existencia del espacio con incΓ³gnitas x, y, z, t, …). En estor pueden existir otro(s) espacio(s) vectorial(es) mΓ‘s pequeΓ±o(s) o igual(es).

El sistema es linealmente independiente si π‘Žπ‘‰βƒ—+π‘π‘ˆβƒ—+…=0 si y solo si π‘Ž=0, 𝑏=0…

Espacios vectoriales

Operaciones con subespacios

In a system of linear equations of the form:

\left\{\begin{matrix}
a_{11}x_1 + a_{12}x_2 +a_{13}x_3 + ... + a_{1n}x_n  = b_1 \\
a_{21}x_1 + a_{22}x_2 +a_{23}x_3 + ... + a_{2n}x_n  = b_2 
\\
 ... \\
a_{m1}x_1 + a_{m2}x_2 +a_{m3}x_3 + ... + a_{mn}x_n  = b_m 
\end{matrix}\right.

where $a_{ij}$ are the coefficients, $b_i$ the known factors, and $x_j the unknowns.

If $βˆ€b = 0$ the system is homogeneous.

A _solution is a n-tuple $(s_1,s_2,...,s_nοΌ‰$ that satisfy all equations.

  • A system is incompatible if it admits no solutions for all equations simultaneously.
\begin{Bmatrix}
x_1 + x_2 = 3 \\
x_1 + x_2 = 1
\end{Bmatrix}
  • A system is said to be compatible if it admits solutions.
    • Two equations are independent if no equation is linear combination of another.
    • If a system of n equations and n unknowns is independent it has only one touple of solutions.
    • If a system has dependent equations or fewer equations than unknowns then it can have an infinite set of solutions.
\begin{Bmatrix}
x_1 + x_2 = 3 \\
2x_1 + 2x_2 = 6
\end{Bmatrix}

Two linear systems are called equivalent if they have the same solutions.

MΓ©todo Gauss

We know that the system of equations can be equivalent (same solutions) to other systems of equations. Gauss method consists in using these simple transformations to find a simpler system with easy-to-find solutions.

We can use the next transformations:

  • Adding two equations
  • Switching two equations
  • Scaling one equation
  • A linear combination of two equations added to an equation

Rango

Inversa

Determinante

Aplicaciones Lineales

wxMaxima

Historia