Understanding Vector Matricies?

[krusty]

I'm having a horrendous amount of trouble understanding Vector Matricies. I think what really stumps me is this:

 

If I have a 3x3 vector like this:

 

1 0 0

0 1 0

0 0 1

 

Or to apply a value to each I would put: (which is probably technically wrong)

 

a d g

b e h

c f h

 

I understand if I have a vector representing a shape on a Cartesian plane and I want to

translate, rotate, refect or something else it.... :

The vector could look like this: (I'm not sure what the 3 would represent)

 

5

2

3

 

......I can multiply or add the 3x3 matrix to it and by changing the values a to h I can effectively translate, rotate, reflect or the matrix to a more or lessor degree. (I have no doubt made technical errors even in my explanation)

 

My question is.... what does each value in the matrix stand for? Is 'a' always a number I would change to translate the shape along the x axis and 'd' a value I would change to move a the shape a along the 'y' axis. I'm having trouble understanding why we even represent these numbers in a matrix form?

 

Is there a really basic explanation out there which could help me understand this? Despite hours of reading on the web, I'm having trouble knowing how to even begin understanding them.

 

Thanks

 

 

 

[Unity+]

From my understanding of vectors and matrices, here is the standard format when doing operations on vectors in matrix form.

 

[math]\begin{bmatrix} A & B &C \\ x_{1} &x_{2} &x_{3} \\ y_{1}& y_{2} &y_{3} \end{bmatrix}[/math]

 

Where the top row are labels of each vector while the other values represent the values in each vector.

 

From here, you can do the proper transformations required.

 

This provides information on those concepts: http://www.mathplanet.com/education/geometry/transformations/transformation-using-matrices

[Sato]

The values in a vector can represent many things, but most likely you're dealing with euclidean vectors, and so you can take them to represent positions in space. Each element then represents some distance along a spatial dimension. For example, you can take the scalar [math]x[/math] to represent the distance along just one dimension, such as length, height, or depth (canonically [math]x[/math] is length). If you have a two element vector, you can take [math]\begin{bmatrix}x \\ y\end{bmatrix}[/math] to represent some magnitude in length, and some magnitude in height; that is, imagine it points to some place on a 2D plane, a little to the left if [math]x[/math] is negative, to the right if [math]x[/math] is positive, or in the centre if [math]x[/math] is 0. Similarly, it'll point lower if [math]y[/math] is negative, in the vertical centre if it's 0, or higher if it's positive. Adding an extra element, [math]z[/math], just adds a description of where it points in an extra dimension; usually, [math]z[/math] will denote the 3D depth, already given [math]x[/math] and [math]y[/math] for 2D length/height; together they, as [math]\begin{bmatrix}x \\ y \\ z\end{bmatrix}[/math], represent some position in 3 dimensions. For example, [math]\begin{bmatrix}5 \\ 3 \\ 1\end{bmatrix}[/math] will point 5 units (inch, cm, metre, etc) to the right, 3 units up, and 1 unit forward. A matrix [math]\begin{bmatrix}x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \\ z_{1} & z_{2} & z_{3}\end{bmatrix}[/math] represents a collection of vectors. Note that you could also represent the matrix as [math]\begin{bmatrix}x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3}\end{bmatrix}[/math].

 

Here's a nice animation illustrating these ideas:

[krusty]

Thanks for replying.

 

You have gone through allot of trouble to answer my question but unfortunately I still don't understand.

 

I probably have to go back to the roots and discover 'why a matrix' and 'how does a transformation matrix relate to the values it is designed to change and move' etc.

 

I'm completely stumped on why we stack the numbers one top of each other in a matrix form. I would assume that we do so because in that form it can be seen how the values in a transformation matrix interact with a vector describing a shape or line on a cartesian plane. It's one thing to know how to do the calculations but to really understand it, I'm not sure how I can get to that point ;(

 

Where do I even start to get an understanding of why we even utilize a matrix?

 

Thanks and sorry to be a big best. I have no doubt others will benefit from this as well.

[timo]

Wrote a long post and then hit a wrong button. I'll try with short bullet points instead:

- Think of matrices as a compact convention how to write down operations performed on vectors.

- The notation is unique: Two operations are the same operation exactly if their matrix is the same.

- Only operations with a certain property (linearity) can be expressed by matrices. That implies that if something is expressed as a matrix you immediately know that these linear properties exist in your problem, which may not be obvious otherwise.

- Linear operations can be combined, which has a compact notation equivalent on matrix level: First applying operation A on a vector x, and then applying operation B on the result has the matrix notation BAx.

- The whole matrix thing gets really interesting when realizing that the combination of two linear operations is a linear operation by itself (the operations form "an algebra", in this case "a ring"). Above combined operation could be written Cx, where C=BA. When B and A are represented by a matrix, there is a direct construction rule telling you how to calculate the matrix of C. In other words, you can do math with operations/matrices as if they were numbers (except that a few rules are different). That is extremely powerful.

- Deliberately expressing something as matrices can be extremely helpful on an advanced level. In advanced math, your main problem often is to understand the mathematical behavior of your system. If parts of it can be expressed in matrix formalism you directly see that matrix math holds. In fact, just yesterday I solved an optimization problem by re-arranging my equations/objects such that they could be expressed as the extremely common equation Ax=b, with x being the sought-for parameters. Methods to solve this equation are well known, and in fact I stopped by work at this point (I was merely looking for a construction algorithm for a solution, so my work was done after getting to a point where well-known standard algorithms can take over).

 

Bottom line: Main point of matrix notation and matrix math is that it appears very often, so knowing to work with matrices can help you in many problems. Be it on the calculation level (knowing how to efficiently perform calculations) or on the mathematical level (knowing abstract mathematical properties of your system because you are familar with the abstract mathematical properties of matrices).

[krusty]

Thanks very much for that explanation. I'm starting to get the gist of it. As a matter of fact, after looking at the very basic forms of liner algebra, my understanding has improved a whole lot more. Thanks for the help (all the help above), if anyone wants to add to this thread please do so.