What is the purpose of Linear Algebra

[matqkks]

How much emphasizes should be on proof on a first course in Linear Algebra?
I sometimes feel that they (proofs) crowd out a coherent vision for linear algebra.
However I also think a central theme of a Linear Algebra course is to learn reasoning even though it does not always succeed.
The audience is first year undergraduate students studying mathematics and physics but maybe extended to engineers. They generally struggle with the idea of proof.

 

[chiro]

How much emphasizes should be on proof on a first course in Linear Algebra?
I sometimes feel that they (proofs) crowd out a coherent vision for linear algebra.
However I also think a central theme of a Linear Algebra course is to learn reasoning even though it does not always succeed.
The audience is first year undergraduate students studying mathematics and physics but maybe extended to engineers. They generally struggle with the idea of proof.

Proofs are pretty central to all of mathematics: one of the reasons we place a lot of trust in mathematics as a whole discipline is because of the proof mechanism. If we couldn't have such a system that people agreed on, the whole thing would come down like a house of cards and we might as well not have mathematics at all.

Linear algebra is more or less the study of representing and analyzing linear spaces (more or less). A linear space is something that acts like a vector or an arrow and the definition is basically that f(X+Y) = f(X) + f(Y) and f(aX) = af(X) for an appropriate object X and Y from some vector space.

The vector space axioms are more or less based on proving the above two identities in a rigorous way and when these are proven, we are gauranteed the linearity characteristics of the space given a field and an addition operation.

Then you get the situation of trying to solve a system of linear relationships and this leads into a whole lot of material including analytic and numeric frameworks for finding solutions. Also there is the topic of finding solutions that are the best given that we have an operator that is singular (no inverse) or close enough to singular and these kinds of things deal with pseudo-inverses.

You also deal with the formal idea of decomposing a linear system into minimal representations and along the way you deal with spanning and basis sets as well as the clarification of what dimension is and how this relates to other properties of linear analysis.

Then like every other area of mathematics, we look at possible decompositions for structures and this ends up indirectly leading to eigen-decompositions. There are also other decompositions but they are part of a later course.

Then on top of the vector space you introduce normed and inner product spaces in which the latter adds geometry to your space. From this you can deal with projections and introduce general decompositions of vector spaces with an inner product.

This kind of thing forms a lot of the stuff dealing with hilbert spaces and integral transforms which concerns itself with things like Fourier analysis and wavelets.

Finally linear operators form the basis of studying general differentiation for multivariable calculus (both differential and integral) and also for tensor theory (generalized co-ordinate system theories) which is pretty useful. Also the differential is a linear operator and this is utilized in more general theories.

You may actually do the simplest case of tensor analysis where the two geometries are completely flat (no curvative at all) when you have to transfer between two coordinate systems (given vector A in C1 find A in C2 where C1 and C2 are constant matrices). The extension to general systems is a lot easier to understand if you have been through this example.

Hopefully that will give you some idea.

 

[Windowmaker]

I had the same feeling as the OP when I had linear algebra. I could do the "problems" just fine. Like find if the system is consistent or find the null or col space, but give me a proof, and I was almost lost. Good thing I'm going into engineering, where there aren't as many proofs.

 

[jbrussell93]

What if the course name is "numerical linear algebra"? Is this less likely to have as many proofs? The course description says that knowledge of mathematica or MATLAB is recommended...

 

[Windowmaker]

Sounds like it would be less proof based. The proofs in linear algebra aren't that bad, considering you know the concepts. For example, to prove if a transformation is linear, show that it is generally closed over addition and scalar multiplication. But if you don't know what "linear" entails, the proof is going to be difficult. I had this class last semester, and it wasn't that bad. Numerical Linear Algebra might be less proof based, if it is, it should be relatively easy. Unless of course its like a 400 level class, but if its a 200 level, it shouldn't be to bad. Just my opinion, so take it with a grain of salt.

 

[jbrussell93]

Sounds like it would be less proof based. The proofs in linear algebra aren't that bad, considering you know the concepts. For example, to prove if a transformation is linear, show that it is generally closed over addition and scalar multiplication. But if you don't know what "linear" entails, the proof is going to be difficult. I had this class last semester, and it wasn't that bad. Numerical Linear Algebra might be less proof based, if it is, it should be relatively easy. Unless of course its like a 400 level class, but if its a 200 level, it shouldn't be to bad. Just my opinion, so take it with a grain of salt.

my school goes by 1000's, and yes it is a 4000 level... Ouch

 

[Windowmaker]

Well I am not to familiar with 4000 level math classes, but I can tell you it will be harder than the linear algebra course I took at my university. What major are you by the way?

 

[jbrussell93]

Biological engineering but this course is not required for the major. I just want to take it because:
- most other schools require it
- many grad schools want to see it
- it will give me a math minor.

 

[Windowmaker]

It shouldn't be to bad, I had a horrible proof, like .5 rating on rate my professor and got a B- and I'm far from a math genius. Just got to put some work in and go to kahn academy a lot.

 

[Windowmaker]

Prof*

 

[jbrussell93]

Well fortunately I checked the grade distribution and out of 20 kids each semester, an average of 2 people per class DIDN'T get an A so either the teacher is good/easy or the kids are geniuses... I'm hoping for the former

 

[Windowmaker]

Where did you find that statistic?

 

[jbrussell93]

Our school has an awesome resource (that most people don't know about) where you type in a professor's name or class name and it tells you how many people got A/B/C/D/F & average GPA for each semester the class has been taught and who was teaching it that semester. It's great for figuring out who to avoid or vice versa.

 

[Windowmaker]

Nice, is there a site or something where I can see if my school has that resource?

 

[Jorriss]

Nice, is there a site or something where I can see if my school has that resource?

Check your schools website. UCSD has something like that. I stumbled across it once.

 

[20Tauri]

What if the course name is "numerical linear algebra"? Is this less likely to have as many proofs? The course description says that knowledge of mathematica or MATLAB is recommended...

I would assume if it has "numerical" in the title that it is less likely to involve proofs, but that's not a hard and fast rule or anything. My college used it as a transition into proof-writing (although there was also a separate, later class that got into more advanced proofs and logic) and we still used Mathematica a lot. It's mainly because things like Gaussian elimination and inverting matrices take a lot of time to do by hand and are only character-building the first couple of times you write out all the steps.