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
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