Posted by: arosenbusch | June 10, 2009

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.

Eisenhower MatrixExample: 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.

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Responses

  1. Hi,

    could you please explain why there are 9*8/2 ways to choose the interval? I don’t get the idea.

    Regards,

    Martin

    • oh, that is because to pick an interval, you pick the start and the end line. A chess board has 8 squares along each side, so there are 9 lines, right? There are 8*9/2 ways to chose 2 out of 9.

      In general there are (k-1)k/2 ways to pick 2 out of k.

      If you do not believe that, you might want to get a feeling for the rule, by seing that there are 3*4/2 ways to pick 2 elements out of a set of 4.

      An easy reference for me to give is http://en.wikipedia.org/wiki/Binomial_coefficient#Combinatorial_interpretation

      However this reference goes a little far.. I believe you will be able to figure out why it is 9*8/2 without knowing anything about binomial coefficients. 🙂

  2. HI,

    thank your very much for your explanation. I was missing that we are about to pick an interval on each axis!

    Regards,
    Martin

    • Well I’m very glad you asked then, because that’s the “independence and dimension” aspect of the example, right? 🙂


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