Ask HN: What should I know about linear Algebra, and where will I use it? | Hacker News Reader

Ask HN: What should I know about linear Algebra, and where will I use it?

7 pointsColinWright11 comments

People say that Linear Algebra is used all over machine learning, and other hot topics, but I rarely see actual case studies, or explanations of actual examples.

I'm pretty sure Linear Algebra isn't used in most web or CRUD applications, so where have you seen it in action?

What should I know about Linear Algebra?

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Google pagerank:

http://www.ams.org/samplings/feature-column/fcarc-pagerank

One of the best answers to "when while I ever use this stuff?" a teacher could provide, even when in high school.

Graphics. You'll see it especially in game development.

If you can comprehend thinking in multi dimensions you should be fine. Obviously you'll need to learn matrix arithmetic.

How does Linear Algebra feature in game development? Can you provide a reference? My Google-fu must be lacking today ...

Thanks.

Say you know the coordinates of all the corners of a cube in three dimensions. If you wanted to move that cube to the side and turn it, you could first add some fixed number to all the x coordinates of every point, then do some trig to find the coordinates again after a rotation. Turns out you can produce the same output with fewer steps using matrix math. That is, you can produce a single matrix which is representative of the transform "move right m units and turn d degrees". And you can apply that transform matrix to any set of points you want.

At least, this is what I've gleaned so far as someone that doesn't really understand. I welcome corrections.

That's the best answer I've had so far, but it seems like it's just talking about matrix operations. There doesn't really seem to be any "proper" Linear Algebra in here. Nothing about bases, change of bases, null spaces, proofs that all bases have the same size, eigen-values and eigen-vectors, none of the juicy stuff. Just matrix multiplication.

So I was wondering if the more advanced stuff is used, or if it really is just the dabbling in the shallows.

Thanks.

I wrote a short introduction to some of the projection math used in 3d graphics, which gets slightly into it, though not into greatly juicy stuff: http://www.kmjn.org/notes/3d_rendering_intro.html

Beyond just matrix arithmetic, a change of basis is usually used in viewing, to change an x-y-z "world" coordinate space into a u-v-w camera-relative coordinate space (discussed in the "View space" section).

The homogeneous coordinates used to encode affine transforms into matrix multiplications also have a rigorous linear-algebra background, but admittedly they can be treated as just a strange hack (use a "dummy" 4th dimension for 3d coordinates) without really needing the theory.

Another example is Vector projection [1], this is how video games (or design software) map 3D worlds onto the 2D screen [2].

[1] http://en.wikipedia.org/wiki/Vector_projection [2] http://en.wikipedia.org/wiki/3D_projection

You need quaternions to avoid gimbal lock. In my small experience with 3D graphics, you would only deal with linear algebra on building cameras.

http://en.wikipedia.org/wiki/Quaternion

I'm using linear algebra in a CRUD app. In short, the app generates a lot of math problems. I use some of the more basic aspects of linear algebra to do this. Matrix equations, etc.. Linear algebra is used extensively in certain types of software like etap, pspice, etc.. Anything where systems can be represented by systems of equations will usually benefit from employing matrix operations.

http://en.wikipedia.org/wiki/Eigenface comes to mind

also http://en.wikipedia.org/wiki/Singular_value_decomposition seems to pop up every couple of months or so in a "Oh they do that with SVD?" way

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