Figurate Number

A figurate number, also (but mostly in texts from the 1500 and 1600s) known as a figural number (Simpson and Weiner 1992, p. 587), is a number that can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a regular polygon, the number is called a polygonal number. The polygonal numbers illustrated above are called triangular, square, pentagonal, and hexagonal numbers, respectively. Figurate numbers can also form other shapes such as centered polygons, L-shapes, three-dimensional solids, etc.

The th regular -polytopic number is given by

			(1)
			(2)
			(3)

where  is the multichoose function,  is a binomial coefficient, and  is a rising factorial. Special cases therefore include the triangular numbers

(4)

tetrahedral numbers

(5)

pentatope numbers

(6)

and so on (Dickson 2005, p. 7).

The following table lists the most common types of figurate numbers.

figurate number	formula
biquadratic number
centered cube number
centered pentagonal number
centered square number
centered triangular number
cubic number
decagonal number
gnomonic number
Haűy octahedral number
Haűy rhombic dodecahedral number
heptagonal number
hex number
heptagonal pyramidal number
hexagonal number
hexagonal pyramidal number
octagonal number
octahedral number
pentagonal number
pentagonal pyramidal number
pentatope number
pronic number
rhombic dodecahedral number
square number
square pyramidal number
stella octangula number
tetrahedral number
triangular number
truncated octahedral number
truncated tetrahedral number

Ref: mathworld.wolfram.com

Linear Algebra

Linear algebra is the study of linear sets of equations and their transformation properties. Linear algebra allows the analysis of rotations in space,least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering. Confusingly, linear algebra is not actually an algebra in the technical sense of the word "algebra" (i.e., a vector space  over a field , and so on).

The matrix and determinant are extremely useful tools of linear algebra. One central problem of linear algebra is the solution of the matrix equation

for . While this can, in theory, be solved using a matrix inverse

other techniques such as Gaussian elimination are numerically more robust.

In addition to being used to describe the study of linear sets of equations, the term "linear algebra" is also used to describe a particular type of algebra. In particular, a linear algebra  over a field  has the structure of a ring with all the usual axioms for an inner addition and an inner multiplication together with distributive laws, therefore giving it more structure than a ring. A linear algebra also admits an outer operation of multiplication by scalars (that are elements of the underlying field ). For example, the set of all linear transformations from a vector space  to itself over a field  forms a linear algebra over . Another example of a linear algebra is the set of all real square matrices over the field  of the real numbers.

Ref: mathworld.wolfram.com