Independence and Dimension

in mathematics, the dimension of a vectorspace is given by the largest possible number of linearly independent vectors in this space.

Whenever different attributes are independent, it can be helpfull to think of them as different dimensions, to visualize them on different axis and to look at the combinations of the attributes in that way.

Example: The Eisenhower Method is a tool for priorization of tasks. It suggests that one thinks about the urgency of each task as independent from its importance. The Eisenhower Matrix is then a fourfold table where the x-axis represents urgency and the y-axis represents importance. This tool is used by leaders all over the world to visualize the difference between urgent and important tasks. If neglected, people tend to do urgent unimportant tasks instead of attacking important issues at an early state. Having the two-by-two matrix helps you to understand, that importance and urgency are independent attributes that should not be confused.

Whenever something has several dimensions, it can be helpfull to think of it as being composed of independent attributes.

Example: How many rectangles are there on a chess board? At first this is a difficult question. There are of corse 64 little squares, and 1 square of size 8×8… but how many other rectangles are there? It is a truely remarkable feat to naively count them all without pen and paper.
Of corse a rectangle is the product of a range on the x-axis and another range on the y-axis. The two ranges can be chosen independently.  Since there are 9*8/2 = 36 ways to chose an interval on each axis, there must be 36*36 = 1296  rectangles on the chess board.

Beware: Not all attributes that are represented multi-dimensionally really are independent. It is possible to show that two attributes are not independent by plotting their instances in a two dimensional manner. Dependence (or correlation) can then be seen, since the instances do not cover the whole two dimensional area. An example is Hans Rosling’s study of life expectancy and income in different countries.