Vectors, Matrices and Determinants. Oh my.

[xxChrisxx]

I never really enjoyed learning the theory of maths and generally tried to avoid it at all costs since leaving University. However I'm looking at learning kinematic and dynamic chassis modelling and it requires extesive use of vectors and their transformations.

I can follow the problems in Stroud - Engineering Mathematics, I'm having trouble relating this to exactly what's going on. As I've forgotten almost everything, it's like I'm starting from scratch learning it for the first time.

Can anyone recommend a good book, or other learning material on this?

Thanks.

 

[chiro]

I never really enjoyed learning the theory of maths and generally tried to avoid it at all costs since leaving University. However I'm looking at learning kinematic and dynamic chassis modelling and it requires extesive use of vectors and their transformations.

I can follow the problems in Stroud - Engineering Mathematics, I'm having trouble relating this to exactly what's going on. As I've forgotten almost everything, it's like I'm starting from scratch learning it for the first time.

Can anyone recommend a good book, or other learning material on this?

Thanks.

Do you want deep understanding like why multiplication is defined, how determinants are defined and so on?

If so I would recommend you read "The Linear Algebra A Graduate Student Ought to Know" 2nd Edition written by Jonathon S. Golan, Published by Springer 2007.

Its a very deep book and maybe it might be deeper than what you want, but nonetheless it is pretty comprehensive. Its not a book that applied scientists and engineers would commonly used, but again if you want deeper understanding I think it might serve you.

 

[xxChrisxx]

To be honest, I'm not really sure what I want. It's got to the point where I just switch off when I see a page full of mathematical notation becuase I only have a superficial understanding.

That's what I'm trying to sort out.

 

[chiro]

To be honest, I'm not really sure what I want. It's got to the point where I just switch off when I see a page full of mathematical notation becuase I only have a superficial understanding.

That's what I'm trying to sort out.

You might get switched off this book, but if you want to get beyond the superficial understanding, the above book should help.

I'll give an example with determinants.

Most courses in linear algebra just introduce determinants, and then formulas with the determinant and then you just use the formula.

The book I mentioned above has a whole chapter on determinants. It starts off by describing what a determinant function actually is (You might be surprised that there isn't just one way of evaluating the determinant, there are many).

So you want to know the idea of the determinant? You have to consider what a major part of linear algebra is all about which is finding solutions to linear systems. So the determinant is brought in and you get axioms that define the determinant in your identity system, as well as what happens with row swaps, row operations and so on. Using these definitions, you end up with a determinant function.

Its like say with vector spaces. All this is about is treating objects like they were straight line arrows. All you are doing is taking an object and making an analogy with an arrow. The vector space framework basically says that if we can represent things with arrows, then we can also decompose our "big" arrows into a whole bunch of "little" arrows. Like say you have 3D space, you can represent any "big" arrow as a linear combination of "little" arrows.

So in the end the whole vector space, inner product space thing is basically saying we got this object that "behaves" like an arrow and geometrically looks like an arrow, and based on this we want to find a way to turn this "big" arrow into little arrows. Now if you're talking about something like say a normal vector in 3D space, it can be pointless. But what about if you have a function? Well there are many different sets of "little" arrows that you can have and each "set" of "little" arrows can tell you some specific property of the function.

For example Fourier series takes a function (big arrow) and spits out frequency information (little arrow). Now there are lots and lots of sets of "little" arrows that represent specific "properties" in some context (like the Fourier example). Mathematicians are exploring this even presently in research and with areas building on infinite dimensional vector spaces like wavelets, there is going to be even more progress with improving the framework to find the right "sets of little arrows" based on some specific criteria.

 

[xxChrisxx]

Cheers. I'll take a look at the book, it sounds like a more in depth explanation will certainly help me.