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Linear algebra (Set (operations (intersection, union, subtraction,…
Linear algebra
Set
illustration
ven diagram
terminology
subsets
empty set
equal sets
operations
intersection
union
subtraction
complement
Cartesian products
properties
commutative
associative
distributive
Algebraic structures
Groups
binary operation
associative
neutral element
∀a∈ G, ∃a’∈ G such that a∗a’ = a’∗a = e
set
contain
binary operations
set
Rings (V, +, •)
+ and • : binary operation
(V, +) commutative group
• associative
distributive
identity element
Fields
is a
exists multiplicative inverse
example
complex number
forms
polar form
canonical form
operations
+
-
*
/
Theorem of equation polynomial power n
Linear mappings
Operate on
Representation as
terminology
kernel
image
eigenvalue and eigenvector
finding
diagonalization
procedure
conditions
meaning
special case
linear transformation
Mapping
Operate on
terminology
images
inverse images
composition
types
Injective (one to one)
Surjective
Bijective
#
#
Inverse
only on
Vector spaces
application
vector space of matrix
matrices
operations
*
Transposition
inverse
existance conditions
Determination
Gauss – Jordan method
determinant
scalar multiplication
+
special forms
Triangular matrices
Diagonal matrices
Unit matrix
echelon form
symmetric
usage
Systems of Linear Equations
theorem of system of linear equation
solve
Cramer’s rule
Gauss Elimination Method
rank
usual space
polynomial coefficients
bases
originate
linear dependence & linear independance
linear combination & span
coordinate
change of bases
definition
operations
Vector addition
Scalar multiplication
axioms
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
change of coordinate
symmetric
process
orthonormal