Math notes / Algebra

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This is more for overview (my own) than for teaching or exercise.

For some more in-depth material to study, see e.g. the http://www.khanacademy.org/


Math on data:

some introduction · areas of statistics
types of data · on random variables, distributions
Virtues and shortcomings of...
on sampling · probability
glossary · references, unsorted
Footnotes on various analyses

Other data analysis, data summarization, learning




Elementary algebra

Elementary algebra or basic algebra are terms that frequently point at the parts of algebra regularly taught in secondary education, typically the more easily understood parts and may be useful to calculus.

To some degree, the concept of basic algebra seems to exist to contrast it with the more complex stuff typically left until university (though basic algebra courses tends to introduce these concepts only explored later).


Algebra assumes knowledge of arithmetic, and introduces some concepts that are central to like

  • the concept of variables
  • how to play with expressions and how they are affected by certain changes,
  • polynomial equations (linear equation, quadratic equation, etc.)
  • factorization, root determination, and such



Linear algebra

Linear algebra studies things like linear spaces (a.k.a. vector spaces) and linear functions (a.k.a. linear maps, linear transforms), systems of linear equations.


...in part because this fairly specific focus has relatively wide application. A lot of said uses relate to, informally, "when a matrix multiplies a vector, it does something meaningful", and a decent area of linear algebra studies the various useful things that you can do.

Vectors and matrices

Notation and conventions

Contents/uses, properties, operations related to properties

Vectors
Matrices

Can hold any tabular sort of data. Many uses are more specific and constrained. In many cases we deal with data as real numbers.


You may sometimes see irregular matrices or sparse matrices, often in computing. In general, matrices are assumed to be regular and non-sparse.



Uses of matrices

Matrices are used for various bookkeeping of nontrivial data, so have many specific(ally named) uses. Including:


In linear algebra

  • representing certain numerical problems,
for example, and commonly, the coefficients of a set of linear equations (each row being one equation)
in part just a data storage thing, but there are some matrix properties/transforms that make sense


  • Transformation matrix [1]
storing linear transforms in matrices
...so that matrix multiplication, typically on coordinate vectors, will apply that transformation to that vector
see e.g. the workings of OpenGL, and various introductions to 3D graphics
Some single-purpose transformation matrix examples:
Rotation matrix - [2]
Shift matrix - http://en.wikipedia.org/wiki/Shift_matrix (ones only on the subdiagonal or superdiagonal)
Shear matrix - [3]
Centering matrix - [4]


In graph theory and such

  • distance matrix - distances between all given points. E.g. used for graphs, but also for other things where there is a sensible metric.
  • adjacency matrix - [5]
  • further matrices assisting graph theory, including degree matrix[6], incidence matrix[7],
  • Similarity matrix [8]
  • Substitution matrix [9]
  • Stochastic matrix -
    • a.k.a. probability matrix, transition matrix, substitution matrix, Markov matrix
    • stores the transitions in a Markov chain
    • [10]


In multivariate analysis, statstical analysis, eigen-analysis

  • Covariance matrix - used in multivariate analysis. Stores the covariance between all input variables [11]


Other

  • Confusion matrix
a visualisation of the performance of a classification algorithm
rows are predicted class, columns are known class
numbers are the fraction of the ca
the closer this is to an identity matrix, the better it performs
  • representing differential equations



Notes:

  • Many are defined in such a way that certain operations are meaningful (though the meaning of operations can obviously vary).
For example, multiplication of graph's adjacency matrix with itself will express connections in as many steps


  • Real vector space ℝn (often ℝ2 or ℝ3 in examples) are quite common in vectors, and common to various matrix uses.
For example, when solving linear equations, you often have row vectors in ℝn, and many (though not all) operations


Operations

Augmenting - an augmented matrix appends columns from two matrices (that have the same amount of rows) Seen in application to systems of linear equations

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Some supporting terms and properties

Operations

Common concepts around vectors and matrices

Space

Linear combination

Span, basis, and subspace

Simultaneous equations / systems of equasions

Linear independence

Rank

Basis

Vector spaces and related concepts

Orthogonality, orthonormality

Angle between two vectors

More complex concepts around here

Eigenvectors and eigenvalues

Eigenvalue algorithms

Power method / power iteration
Deflation Method

Eigendecomposition

Applications

SVD

On the decomposed matrices' sizes

Definition / main properties

In more detail

Further properties, observations, and uses

See also

Others

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Abstract algebra

Abstract algebra studies the possible generalizations within algebra.

It concerns concepts like group theory, rings, fields, modules, vector spaces, and their interrelations.


Combinations, permutations

Polynomials

Inverse of a function