# Difference between revisions of "Math notes / Algebra"

 This is more for overview of my own than for teaching or exercise. Arithmetic · 'elementary mathematics' and similar concepts Set theory, Category theory Geometry and its relatives · Topology Elementary algebra - Linear algebra - Abstract algebra Calculus and analysis Logic Semi-sorted  : Information theory · Number theory · Decision theory, game theory · Recreational mathematics · Dynamical systems · Unsorted or hard to sort Math on data: Statistics as a field 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 Data massage Data clustering · Dimensionality reduction · Fuzzy coding, decisions, learning · Optimization theory, control theory Connectionism, neural nets · Evolutionary computing

# 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

#### Contents/uses, properties, operations related to properties

##### 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 
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 - 
Shift matrix - http://en.wikipedia.org/wiki/Shift_matrix (ones only on the subdiagonal or superdiagonal)
Shear matrix - 
Centering matrix - 

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.
• further matrices assisting graph theory, including degree matrix, incidence matrix,
• Similarity matrix 
• Substitution matrix 
• Stochastic matrix -
• a.k.a. probability matrix, transition matrix, substitution matrix, Markov matrix
• stores the transitions in a Markov chain
• 

In multivariate analysis, statstical analysis, eigen-analysis

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

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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# Abstract algebra

Abstract algebra studies the possible generalizations within algebra.

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