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| Vectors and Matrices: A Primer |
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Stoffel Member since: 12/2/1999 From: San Diego, USA |
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 Posted - 6/5/2002 1:34:41 PM |
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| Fine article. I'd like to add something about matrices that I didn't learn until far too late (IMHO): Some vector and matrix operations have physical interpretations, which I find very helpful for understanding what's going on. For instace, I think most of us know that the dot product is a projection operation--the magnitude of what vectors A and B have in common is their dot product, which is why orthogonal vectors have dot product = 0. But what the heck is the determinant? Turns out that the determinant is the "volume" of a matrix. This can be proven for 2-D matrices directly (area of parallelogram) and it's only a small stretch to do it for 3-D matrices (using the A dot B cross C, if you really think about what's going on). But this is also true for N-dimensional matrices. This also shows why underdetermined (non-full-rank) matrices have such degenerative properties--they have no volume! It also explains why so many linear algebra formulations divide by the determinant--it's the scaling factor for the volume of the matrix. [edited by - Stoffel on June 5, 2002 5:57:18 PM] |
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python_regious Member since: 2/28/2001 From: Bath, United Kingdom |
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Posted - 6/5/2002 1:44:43 PM |
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I didn't know that about determinants... it sorta makes sense now you've said it.. Thanks for that  Death of one is a tragedy, death of a million is just a statistic. |
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Outworlder Member since: 3/3/2000 From: Fortaleza, Brazil |
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Posted - 6/5/2002 8:08:47 PM |
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| I've always been annoyed by those determinants. I understood when to use them, but not what the heck they were (I mean, they just couldn't discovered all those properties out of thin air). Thanks for the explanation. |
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GBGames Member since: 1/25/2001 From: Chicago, IL, United States |
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Posted - 6/5/2002 9:49:26 PM |
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| I have the Angel book mentioned in the conclusion, and while it is nice supplemental material, it does not explain the ideas behind a few things clearly enough on its own. It was the textbook in my Computer Graphic Design class, and I know that I barely touched it. Just my thoughts. To be honest, OpenGL Game Programming was more helpful to me in that class than the text book. And the Red Book didn't hurt either B-) ------------------------- (Gorgeous graphics)+(beautiful sound effects)+(symphonic music)+(no gameplay) != Good game |
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JeffKershner Member since: 11/14/2001 From: USA |
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Posted - 6/6/2002 8:08:37 PM |
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| The determinant is really the combination of all permutations of a matrix, and that is why is seems like the volume of a matrix. |
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Anonymous Poster |
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 Posted - 11/18/2002 8:19:58 AM |
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| I just would like to thank you Phil ... good job |
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python_regious Member since: 2/28/2001 From: Bath, United Kingdom |
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 Posted - 12/23/2002 6:23:48 PM |
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| Cheers Death of one is a tragedy, death of a million is just a statistic.  |
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Jimfing Member since: 3/22/2002 From: United Kingdom |
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Posted - 2/12/2006 5:58:44 AM |
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| Hi - these is a problem viewing this article using the new GD.net theme (with a white background). The images assume they are on a dark background (like the classic GD.net theme). Thanks! |
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