4. Algebra - JulTob/Mathematics GitHub Wiki
βοΈ Algebra
π The Language of Abstraction and the study of abstract structures. How mathematical models are organized. From the Arabic $\color{#18f}AlβJabr$ ("reallocation")
πͺ¬ Algebra uses $\color{#189}symbols$ to represent entities, reducing our focus into one specific aspect we care about.
π« Equations & Expressions
- π« The art of balance.
β¨ Polynomials & Factorization
- The secret structure of numbers.
βοΈ 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
- Constants:
π
,π
,π
,n
- Variables:
π₯
,π¦
,π§
,π
,π‘
,π
,π
- Numbers:
- Quantities
- Operations
- Addition
οΌ
,β
- Subtraction
οΌ
- Multiplication
Γ
,β
,(π)(π)
,π(π)
,ππ
- Division
Γ·
,βΆ
,πβπ
,πβ£π
- Exponents
πΒ²
,πα΅
,^
,**
- Roots
Βͺβ
,β
,Β³β
- Addition
- Relationship
- Equality and inequality
=
,β
,οΌ
,β€
,οΌ
,β₯
,β
- Proportionality
β
,β·
- Order
βΊ
,β»
,βΎ
,βΏ
,β
,β
- Equality and inequality
- Organization and order
( )
,[ ]
,β¨ β©
,β§β§
,β§β§
,Diagrams β₯β’ββ«
Expressions
Sets of symbols that represent a concept.
bh/2
2ππ
sinπ
Equation & Inequality
Relationships between Expressions.
β―^2ππ - 1 = 0
-1 β€ cosπ β€ 1
π(x) β β―Λ£
Linear Equation
ππ₯οΌποΌπΆ
π₯ οΌ -π/π
Quadratic Ecuations
ππ₯Β²οΌππ₯οΌποΌπΆ
-πΒ±βπΒ²-4ππ]
π₯ οΌ ββββββββββββ
2π
Polynomials
Generalizing, the simplest type of functions are Polynomials.
π(π₯)= πβπ₯βΏ .+. ππ₯Β²οΌππ₯οΌπ
= βπβΒ·π₯βΏ
= π¨ β(π₯-πΌβ)
πΌβ : 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
- Parentheses or brackets
- Exponents, Roots, absolute (left to right)
- Multiplication and division (left to right)
- 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, β¦). En estor pueden existir otro(s) espacio(s) vectorial(es) mΓ‘s pequeΓ±o(s) o igual(es).
El sistema es linealmente independiente si
ππβ+ππβ+β¦=0
si y solo si π=0, π=0β¦
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 $βb = 0$ the system is homogeneous.
A _solution is a n-tuple $οΌs_1,s_2,...,s_nοΌ$ 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