Hacker News Reader

Comment by mdtusz | Hacker News Reader

> The first column of the rotation matrix is the new x-axis expressed in the old coordinate system, the second column is the y-axis and so on. An identity matrix would yield in no rotation since all unit vectors would be the same as the previous coordinate system.

It's a shame that this isn't made more clear in most tutorials and classes. The idea of axis application order really does make things more confusing than is needed.

View on HN

I remember that in my freshman's year, my linear algebra teacher explained that every matrix (i.e., not only rotation matrices) has this property: each column is the result of the operation on each basis element in the old coordinate system. This is true even for rectangular matrices.

Once you get this, you can start building matrices for any (finite dimensional) 'vector space' not just the standard 'vectors as a list of numbers'.

All you need to do is to pick a basis for your vector space, and you can start to represent linear transformations as matrices w.r.t. that basis. Even cooler is when you let go of specific basis choices, and start talking about properties of linear transformations that do not depend on a choice of basis. Things like a determinant, eigenvalues, etc.

I've spent the last year correcting for the poor linear algebra taught in my undergrad. The gaps and inconsistencies left by those classes are a lot harder to repair when you've got a full-time job and kids.

What do you suggest as a good text for learning it right the first time?

I actually suggest finding a book about a topic that actually makes use of linear algebra , since the intro chapters will actually explain the core concepts in a way that is actually useful.

For me, linear algebra didn't click until I read "Numerical Linear Algebra" by Trefethen & Bau. The first 4-5 sections have great explanations of matrix vector operations.

Other than that, the most popular linear algebra book that actually gets things right is "linear algebra done right" by Sheldon Axler.

Edit: ah, forgive me, didn't notice you said "first time". In that case the Axler book is a safe bet, just be diligent doing the exercises.

I found Gilbert Strang's MIT 18.06 lectures on youtube and associated course materials very useful.

I had an incredibly difficult time understanding his linear algebra textbook until I watched his lecture videos.

Yeah, his lectures are very well done. Starts with basics and builds up nicely.

Reply on news.ycombinator.com