Question

What should I look first and stick in my head once and for ever when starting studing Linear Algebra? Is Analitical Geometry really very important? And what else of it, apart from vectors, is used in L.A.?
Thanks in advance and, please, help that curious newbie here ;)

Answer 1

Hmm, this is an interesting question. First and foremost, LA is done differently in every school (even those using the same textbook). This is VERY important to note. For example you could probably start at lecture 30 here and wouldn't be too lost (given an understanding of functions). If you are studying in parallel to some school, then it is necessary to follow the syllabus. Unless you are non-conformist like me (it hurts marks).

Geometry is useful for intuition in the topic. You see, LA will extend far beyond 3 dimensions because it doesn't assume properties of 3 dimensions. You only need to know enough geometry to figure out the distance between 2 {lines/planes/points}. At most, you would need a little more for the understanding of the cross product, but that is seldom taught as part of LA, usually more of vector calculus.

Hmm, LA is actually quite an abstract subject. It helps to pay attention to the applications and the ideas behind the properties which allow such an application. Matrices are artificial constructs with certain properties which make them useful. In particular, like equations, manipulating them preserves certain initial constraints. For example, you could do stuff from finding a formula for a Fibonacci numbers (which is a linear recurrence) to graphics transformations, least-squares fitting of curves to data, graph theory, etc (well, it all starts from solving simultaneous equations). Behind each of this there is an idea, and this idea is important.

You don't have to cover every topic, you just need to get used to the idea of a mathematical object - you can use it for any purpose you can force it to do within the constraints. That said, this means that keeping the order given in the MIT lectures is a good idea. The first 9-10 videos are foundation. The rest are topics, some more important than others. I would argue keeping 14-22 (but this is my opinion). The rest are cool ideas.

Of course having said the above, some mathematicians would kill me for being too practical =)

Answer 2

Applied math rocks!

If you like the MIT OCW lecture format and Prof Strang's lectures in particular as much as I do, you should check out his grad courses on boundary value problems and initial value problems.

http://ocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-fall-2008/

http://ocw.mit.edu/courses/mathematics/18-086-mathematical-methods-for-engineers-ii-spring-2006/

Enjoy!