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From Odwiki

If anyone happens to read this by accident, please ignore it for now. Testing a bit before putting up the *real thing*.

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This chapter on odwiki will deal with the essentials of Vector Algebra. However, Vector Algebra is just a special area of Linear Algebra, and although it might be the most common form of Linear Algebra we encounter in Houdini, it is nevertheless worth spending some time introducing the more general Linear Algebra. Later on, once we've covered Vector algebra, we'll see that many of the same concepts can be applied either directly, or with only slight modifications, to fairly different areas.
For those of you who don't wish to read this small introduction on Linear Algebra, feel free to skip ahead to Vector Algebra.

Contents 1 Linear Algebra 1.1 Introduction 1.2 Fields of Application 1.3 Literature and Resources for further reading 2 Vector Algebra 2.1 Euclidian space 2.2 Points and Vectors - similar yet different 2.3 Forumla test

Linear Algebra

What is Linear Algebra? Well, as with all abstract terms, there are many definitions. I like to think of Linear Algebra as a language that allows us to precisely describe both the problem and the solution to common problems in mathematics and sciences. The importance of having such a tool should be self-evident, as without a proper description of the problem there would be no way for us to understand it, let alone solve it. The beauty of Linear Algebra is that it allows us to describe these problems at a very general level, and find solutions to them without having to worry too much about the details. This beauty comes at a price however: it is all very abstract and somewhat hard to grasp at first.

Introduction

'Modern' Linear Algebra is a comparatively young field in mathematics, but it has evolved very quickly; to the point where it is now a central part of mathematics. It all started with the definition of vectors in two- and three-dimentional spaces; Vectors are a central key to Linear Algebra. The concept of vectors has since been extended to include spaces of arbitrary, or even infinite dimension. The classic idea of a vector is that of a directed line segment with an associated length and direction. However, the broader modern view of vectors includes polynomials, matrices, and functions. In these so called vector or linear spaces, we work with linear operators which, without going into any details right now, have the important property of ensuring that our vectors always end up in a linear vector space, even after transformation.

• • Linear Transformation
    1. Properties

Fields of Application

Here you can find a list of fields where linear Algebra plays an essential part. It's more meant as an appetizer and don't be worried if not everything is clear yet.

• Image Compression: There are basically two types of image compression so far: lossless and lossy image compression. The current standard for image compression is still JPEG and key idea is the overservation that the eye doesn't notice very slight changes in color variation. With the help of a basisfunction the image gets transformed and the value range of the image decreases. As the range of possible values for each pixel gets decreased we can store the image in a smaller file. Unfortunally the JPEG transformation is not lossless and hence no linear transformation (i.e. we cannot reconstruct the original file from the compressed image file, it's not a bijective mapping).
  People haven't stopped there and eventually the jpeg standard will be replaced by the new jpeg2000 compression. Here a different basisfunction is used: wavelets. The idea is again in reducing the amount of data and transforming the image in regions that contain data that is 'important' and parts that can be emitted without great loss of visual accuracy. The major advantage of wavelets is that they not only have a locality in the spectral frequency but as well in the time domain, compared to the discreete cosinus transformation (DCT) used for jpeg compression or the fast Fourrier transfromation (FFT) that is widely used as well. As a result of these properties of wavelets we can even do a losloss compression with wavelets with decent compression. This new format has many other advantages, but more in depth on this and the math behind, once we mastered the basics of Linear Algebra. For now, just keep in mind that almost all discrete signal compressions (e.g. audio, video, images) make use of linear algebra techniques.
  For a detailed description of the jpeg2000 compression format look here [1]

• Tracking and 3d reconstruction from 2d image sequences:
  
  

• Solving Differential Equation Systems:

Literature and Resources for further reading

Vector Algebra

In this section we will worry only about our familiar tree-dimensional Eucledian space.

Euclidian space

We will use the symbol E³ to denote the three-dimensional Euclidian space. This is the one we're all that used to and usually associate with 3d-space. In math terms an Eucledean space is a set whose elements satisfy the five axioms of Euclid. We'll soon see what these are. Most of us associate the Cartisian coordinate frame with the Euclidean space, however it's important to note that the three-dimensional Euclidian space can be represented by the Cartesian frame, but it's only a representation. There are quite a few other representations and luckily they won't change the space and if you think about it, it's rather logic since we all 'live' in the Eucledian space, but the Cartisian frame is a man-made construct so we can describe things in Eucledian space. Through such an assignment we establish a one-to-one correspondence between E³ and R³ and this allows us to talk about points with their according coordinates as if they were the same, though strictly they aren't.

Points and Vectors - similar yet different

Forumla test

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