Math

 

Basic Linear Algebra

  1. Coordinate Systems
  2. Points and Vectors
  3. Vector Products
  4. Summary Sheet

Linear Algebra

The algorithms we will present only use basic linear algebra. This will include points and vectors, but not matrices. However, we occasionally use 2x2 and 3x3 determinants to write some formulas,

Notation

This book uses the following conventions for notation.

Scalars

lower case italic

 a,b...x,y...

Points

UPPER CASE ITALIC

 P, Q, R, V, ...
P=(p1,p2,...pn)=(pi)

Vectors

lower case bold

 a, b, ..., u, v, w, ...
v=(v1...vn)=(vi)

Geometric Objects

UPPER CASE BOLD

L, R, S, T, ..., P, ...
DELTA, OMEGA, PSI, THETA-cap-bold, ..., DELTA-P0P1P2, ...

Functions

Regular font

A(OMEGA), f(x), ...

 

Font of the result

P(t)=(x(t),y(t)), v(x,y), ...

Coordinate Systems

Points are positions in space. When first introduced in Greek geometry, there was no formal method for measuring where a point was located.  Points were primitive entities. Much later in history, in the early seventeenth century, Fermat and Descartes introduced the idea of using a linear coordinate system to specify point locations, using algebraic equations to describe geometric objects (such as lines), and using algebra to solve geometric problems (such as computing the intersection point of two lines). We will only use rectangular linear coordinates, and not other methods for referencing points, such as polar or spherical coordinates

A (rectangular linear) coordinate system has an origin as an absolute reference point whose location is fixed, and non-zero axes as a set of direction vectors that determine the directions in which to make measurements. A coordinate system can specify points in n-dimensional space as a ordered n-tuple of numbers (x1,...,xn) and the numeric rule: start at the origin, go distance x1 in the direction of the first axis, stop, now go distance x2 in the direction of the second axis, stop, and so on until done with xn. The final location that one reaches is the point specified. This is much like finding the treasure on a pirate's map; for example, start at the old oak tree as the origin, first go east 10 paces, second go north 20 paces, and third dig down 3 feet to locate the treasure. The ordered set of numbers (x1,...,xn) is called the coordinate of the point it specifies. And a coordinate system is said to span the set of points that it can specify. The set of all these points is called the coordinate space. A specific set of axes spanning a coordinate space is called a coordinate frame of reference or a basis for the space. Clearly, different coordinate systems can span the same coordinate space, just as the pirate may have used any of several origin trees or direction axes for his treasure map.

The axes are said to be independent if they do not depend on each other, meaning that none of them can written as a combination of the others. When independent, they are a minimal set of axes for the coordinate space, and the number of axes in the set is the dimension of the space.

 

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