Linear algebra

Set

illustration

ven diagram

terminology

subsets

empty set

equal sets

operations

intersection

union

subtraction

complement

properties

commutative

associative

distributive

Cartesian products

Mapping Operate on

terminology

images

inverse images

composition

types

Injective (one to one)

Surjective

Bijective # #

Inverse only on

Algebraic structures

Groups

contain

binary operations

set

binary operation

associative

neutral element

∀a∈ G, ∃a’∈ G such that a∗a’ = a’∗a = e

Rings (V, +, •)

set

+ and • : binary operation

(V, +) commutative group

• associative

distributive

identity element

Fields is a

exists multiplicative inverse

example

complex number

forms

polar form

operations

+

-

*

/

Theorem of equation polynomial power n

Vector spaces

application

vector space of matrix

matrices

operations

special forms

Triangular matrices

Diagonal matrices

Unit matrix

usage

Systems of Linear Equations

echelon form

symmetric

*

bases

originate

linear dependence & linear independance

Transposition

rank

theorem of system of linear equation

inverse

existance conditions

Determination

Gauss – Jordan method

solve

coordinate

change of bases

Linear mappings Operate on Representation as

terminology

kernel

eigenvalue and eigenvector

diagonalization

procedure

special case

linear transformation

Euclidean Spaces is an

Inner product spaces

Length (or Norm) of vectors

Orthogonality

Gram-Schmidt orthonormalization process

Projection and least square approximations

Orthogonal matrices

orthogonal diagonalization

application

Quadratic form

transformation to canonical form

quadratic line

quadratic surface

canonical form

Cramer’s rule

definition

operations

Vector addition

Scalar multiplication

axioms

scalar multiplication

usual space

+

polynomial coefficients

Gauss Elimination Method

determinant

image

finding

conditions

linear combination & span

meaning

orthonormal

change of coordinate

symmetric

process

Operate ononly onis aOperate onRepresentation asis an