Could somebody help motivate linear algebra for me?

[Lanza52]

I'm in a beginning LA course in college. And I just don't get it. Not because it's challenging, but because so far there has been no rhyme or reason to what is happening in this class. To clarify, I have like a 110% in the course, but I feel like I've done nothing then rote learning and regurgitating what I read in the book on my exams.

I'm taking an exam tomorrow on subspaces. I've learned the rules and memorized what to do. But I can't figure out what the hell the rational is for doing this.

What reason could there be redefine vector addition or scalar multiplication? Why should I ever have to test if U+V=V+U? What the hell is an arbitrary nonempty set of objects where 3 * (U, V) = (0, V)?

The only answer I've gotten so far is that "you will use it for such and such in the future." This is about as unsatisfying an answer as when my father used to say "Do as I say, not as I do."

I personally enjoy understanding what is happening and being able to derive what is needed. I don't memorize formulas. If I need a formula, I'm the type that takes the characteristic equation of what is happening and derive what needs to be answered. And this class basically forbids understanding. From everything I've been taught so far, there is nothing to understand and there is nothing to derive, there is just "shove the square block into the square hole."

 

[mathwonk]

learn some differential equations. if L is a linear differential operator, then the solutions of Lf = 0 form a subspace of the vector space of all differentiable functions.

 

[sponsoredwalk]

The only answer I've gotten so far is that "you will use it for such and such in the future." This is about as unsatisfying an answer as when my father used to say "Do as I say, not as I do."

I agree, but get used to it - it's quite common as a response in some form or another I'm
afraid to say. Not always, you'll be lucky to get some great & very instructive help but not
always, the good thing is that there is usually some deep thing behind the question you're
asking so just keep at it regardless Really though, I've gotten this as a response
a lot of times over the years & it's daylight robbery. I really mean it, just ignore that
arrogantly & continue to search until you get a satisfying answer.

What reason could there be redefine vector addition or scalar multiplication? Why should I ever have to test if U+V=V+U? What the hell is an arbitrary nonempty set of objects where 3 * (U, V) = (0, V)?

I'm far too drunk to write up a whole description of relations and functions from the basics
of basic set theory so if you really want to learn about this stuff I urge you to find out what
a relation, a function, a structure & an algebraic structure are if you really some amazing
insight into why u + v = v + u really makes sense but at a basic level if I define an
operation Δ as follows, u Δ v = (u/v) where the right hand side, (u/v) is just ordinary
division, then you'll see why this needs to be taken as an axiom. Okay, so just like my
Δ symbol, + is similar, it's just an operation on u & v. It's made far clearer with a bit of
good notation, ƒ : ℝ x ℝ → ℝ such that Δ : (u,v) ↦ Δ(u,v) = u Δ v. Think about normal
addition, + : ℝ x ℝ → ℝ such that + : (u,v) ↦ +(u,v) = u + v, this is clear notation
that explains what's going on so if we have Δ : (u,v) ↦ Δ(u,v) = u Δ v = (u/v) you can
see that Δ : (v,u) ↦ Δ(v,u) = v Δ u = (v/u) is different! With + it doesn't matter what
order you do it in. In a vector space is that all you need is u Δ v = v Δ u etc... for all
axioms, no matter what Δ actually does to u & v. If any of the notation is unclear to you
a good discrete math or "proof" course or just some study on set theory & functions will
clean it up for you.

I highly recommend watching the linear algebra videos in this course:
http://nptel.iitm.ac.in/video.php?courseId=1097
You can skip the matrix & determinant videos for now if you think you're ready & just
get into the vector space stuff from videos 20 on (ignore the complex variables videos!).
These videos will give explicit examples of operations other than + & where they fail
etc... There's also a great discrete math course on there (and a heck of a lot more on
other topics too!).

I hope there are no glaring errors there, if there are my sober brethren will hopefully clean
them up pour moi Actually, my (u/v) example has an error contained in it if you
think about it for a second but I still get my point across using it

 

[Deveno]

my girlfriend hates math. she really does. why do i bring this up?

i was trying to explain to her the other day, about a very simple puzzle i read in a book. here it is:

a farmer went to his barn, where he keeps his duck and cows. he counted 7 heads, and 18 feet. how many of each animal does he have?

now this really is very simple. no one would likely actually use linear algebra to solve it. but you CAN:

D+C = 7
2D + 4C = 18, so we can form the linear system:

A(x,y) = (7,18), with augmented matrix:

[1 1 | 7.]
[2 4 |18]

row-reduction gives the equivalent system:

[1 0 | 5]
[0 1 | 2], in other words, the farmer has 5 ducks and 2 cows.

ok, so this is pretty basic. but if we have 100 variables, and 100 equations, that's a different story.

in the real (and mathematical) world, often several variables occur togther, that are independent of each other. now, often, the equations that govern their behavior are complicated. but the good news is, we can still use their behavior "right now" to make a working model of some larger "state space".

that's where vectors come in. vectors are a very useful concept. a lot of different things can be vectors: colors, polynomials, forces in the physical world, "state space" of deterministic or probabilistic systems, points in space, families of functions. and the bewildering array of their possible interactions can be looked at "one variable at a time", where the intricate, nonintuitve structure can be seen "in a slice view", where the math boils down to the arithmetic we've known since grade school.

so you have vectors. and maps that take vectors to other vectors, without ruining their lovely vector-ness, have a special name, linear transformations. linear transformations are easy to get along with: put 'em inside the plus sign, outside the plus sign, they don't care. they behave very nicely, unlike their unruly cousins the permutations, who never went to commutativity school.

you want to have these linear map thingy-s in your corner. even if a relationship isn't linear, odds are good there's a linear approximation, under suitable restrictions. often, one studies the linear case, before the non-linear case, because we can actually SOLVE the linear case.

if you study enough math, or physics, or even chemistry, eventually you will come across things where A+B and B+A *aren't* the same (although people tend not to use the + sign for those). you should be happy that vectors and matrices ARE that way, they're the "good guys".

eventually, if you don't kill your linear algebra instructor(s) anytime soon, you'll come to the famed "Hilbert spaces" (don't wake Mr. Hilbert...he's dead, and i hear he has a temper...speak of them reverentially), and perhaps feel a certain wistfulness for the days when vectors had finite bases, and proving orthogonality didn't involve complicated integrals and arcane limits. linear algebra is a powerful tool for studying many concepts: economics, gambling, the weather (?!?), differential equations (sorry, I'm all out of wronskians, get your own, buddy), not to mention the many applications in engineering and oh yeah, math. (did you know that steel workers strike a beam to test its soundness by listening to it? perhaps you did, but did you know that eigenvectors explain why that works?).

it's not all just "abstract" either. for example, computer-controlled machining equipment uses some kind of coordinate system to move the machine parts. perhaps one such movement, is the rotation of an arm. somewhere, buried in the machine-code, is the digital equivalent of a rotation matrix (a change of basis). somewhere, someone had to program that in, and i'll bet you $50 they knew linear algebra. game designers use 4x4 real matrices to effect 3-dimensional spatial rotations in their software (so does NASA).

if nothing else, you can keep track of your livestock, by counting heads and feet :P

 

[HallsofIvy]

In essence, Linear Algebra encapsulates the whole idea of "linearity"- any complex problem which we can break down into simpler problems, solve each of the simpler problems, then combine those solutions to a solution to the orignal problem is "linear". That is the basic idea behind writing a vector in terms of "basis vectors" of a subspace.

 

[batboio]

I felt exactly the same way when I started my first year in university and we started studying linear algebra and mathematical analysis. The latter seemed more useful cause at least derivatives and integrals are used everywhere. But the linear algebra seemed really abstract and useless at first.

For example the vector space although abstract in it's definition is a structure which you come upon really often and that's why people defined it with the most basic properties so it can be easily recognizible. That holds true for all the math structures (groups, rings, algebras, etc.). The thing is that the elements of the space may not only be vectors (the vectors you know from school which represent the force in mechanics for example) but also functions, tensors, etc. And once you proove some properties of an element of the vector space coming only from the definition, you can use those for vectors, tensors, functions, etc.

You will start to use the stuff you learned eventually without even thinking about it. If you need examples here are some: electrodynamics (which uses tensor analysis heavily), the special (tensor analysis again) and the general (differential geometry) theories of relativity, quantom mechanics (functional analysis). All of these need LA as a basis (I mean really - the lowest level).

I don't know if your course is proof based or you just learn the algorithms for solving problems but if you want a deeper understanding you need proofs. Also try to imagine the stuff you read about as much as possible cause otherwise you forget more easily.

Hope I've helped and good luck.

 

[Stephen Tashi]

We should also mention that the matrices of linear algebra "are" linear transformations (in the sense that it is always possible to imagine that they are) and linear transformations include those transformations that one would use computing the coordinates of objects moving in space. This would be relevant to real world situations such as computer graphics, planning the motions of robots and, in general, representing the location and orientation of things in space.

 

[lavinia]

I'm in a beginning LA course in college. And I just don't get it. Not because it's challenging, but because so far there has been no rhyme or reason to what is happening in this class. To clarify, I have like a 110% in the course, but I feel like I've done nothing then rote learning and regurgitating what I read in the book on my exams.

What reason could there be redefine vector addition or scalar multiplication? Why should I ever have to test if U+V=V+U? What the hell is an arbitrary nonempty set of objects where 3 * (U, V) = (0, V)?

The only answer I've gotten so far is that "you will use it for such and such in the future." This is about as unsatisfying an answer as when my father used to say "Do as I say, not as I do."

I personally enjoy understanding what is happening and being able to derive what is needed. I don't memorize formulas. If I need a formula, I'm the type that takes the characteristic equation of what is happening and derive what needs to be answered. And this class basically forbids understanding. From everything I've been taught so far, there is nothing to understand and there is nothing to derive, there is just "shove the square block into the square hole."

I don't blame you for being upset. It seems that college math courses are often rote.

The temptation is to tell you all of the applications of linear algebra - which is what makes it necessary - but I won't do this.

Linear algebra is the study of spaces that look like Cartesian x,y coordinates but with an arbitrary number of dimensions. The structure of these spaces is characterized by the ability to change scale uniformly (scalar multiplication) and to form a parallelogram out of two vectors and find its far corner (vector addition).

Interestingly these ideas do not require any idea of distance or angle measurement. They are purely algebraic. This to me is why the subject appears abstract at first.

So what can one study about such simple abstract objects?

Perhaps the most important thing is the way that linear transformations of the vector space into itself move the vectors around. Since linear transformation respect the vector space structure by mapping scalar multiples into scalar multiples and parallelograms into parallelograms, they move entire sub vector spaces into other sub vector spaces and can be understood by characterizing how they do this. Astonishingly, there is a single theorem for finite dimensional vector spaces that totally solves this problem. This theorem is a special case of a more general algebraic theorem that is called the structure theorem for modules of finite type over a principal ideal domain.

In my opinion, this theorem provides a way of putting together much of the subject under a unifying idea. I would be glad to go through it with you if you like.

With any vector space, there is a naturally defined set of other vector spaces. these other vector spaces comes from multilinear maps rather than just plain old linear maps. this is the theory of tensors.

Sadly, linear algebra has a lot of computation but there is really no way around it. Much of this comes from finding invariant subspaces under endomorphisms, representing a linear transformation in terms of a specific coordinates systems, or solving for the intersection of a finite collection of subspaces ( the theory of linear equations).