Is Linear Algebra Always This Confusing?

[cdotter]

It's almost the end of the semester for my first linear algebra course. The course has been taught from a pure mathematics standpoint and I can safely say I have no intuition for the subject. There have no been no physical interpretations or even geometric extensions given in my textbook or lectures.



Am I alone in being clueless and unsatisfied? Is it always like this for the average student?

 

[capandbells]

I think this is normal. Just like geometrical and physics intuition, an intuitive grasp of abstract mathematical objects is something that needs to be developed with lots of work. The unfortunate part is that most people are never given an opportunity to develop this intuition until they are thrown into their first pure math course, which could be overwhelming and turn them away. Stick with it. You may never be able to relate some things in mathematics to familiar physical or geometric concepts (although in some cases you will), but you can develop a taste for the math itself.

 

[Stephen Tashi]

Some linear algebra books are written to give the student a "gentle" introduction to the subject and they concentrate on systems of linear equations, determinants and similar topics that look familiar to students of secondary school algebra. If you have such a text, it isn't unusual to have no intuition about the purpose and nature of eigenvalues, characteristic polynomials, cannonical forms. Even if you understand vectors as used in elementary physics, those topics may be inscrutable.

Linear Algebra is sometimes the first place students encounter the systematic use of mathematical logic and rigorous proofs. Part of the sensation that you feel might be a reaction to that material, which would happen in any course where "real" mathematical reasoning is introduced.

This is a cheap, thin and easy book that gives a geometric interpretation of some topics in linear algebra:
https://www.amazon.com/dp/0486636348/?tag=pfamazon01-20

 

[mathwonk]

try the notes on my webpage.

 

[Geezer]

My comment may not be helpful at all, but I absolutely LOVED linear algebra. If I could have majored in linear algebra as an undergrad, I would have.

I think, as with most anything, it depends on the student. I'm a very visual person with strong spatial skills...I can see the basis vectors, rotations, etc., floating around me. I'm even more comfortable with geometry than linear algebra.

But there are so many other things that just don't come as naturally to me. So, I'm good at linear algebra, but poor in other areas. You might struggle with linear algebra, but you'd surpass me in other subjects.

 

[flyingpig]

What do you mean no geometry? We have Span

 

[wisvuze]

I find it interesting that you can kind of develop an "intuition" for linear algebra by really understanding the notion and symbols, and then the "picture" you have in your head might not be a geometric one.. but a "symbolic" one.
well, Russell did once say that the essence of a good notation is the notations ability to convey its meaning/concept (something along these lines, and it's definitely true)

 

[bennyska]

when you say "pure mathematics," what do you mean? do you mean proof based, doing things in the general sense, or just learning the lower level stuff, just with no real applications? linear alg was one of my favorite classes, on both levels. but it may have helped that the lower level class i took had a pretty easy book that helped explain things, like what this stuff is actually used for. my teacher loved pointing out that computer games use this stuff a lot, any time a 3d picture is shown, involving things rotating, or in a first person-shooter, for example, everything is stretched out depending on what angle you're looking at, which comes directly from this stuff.
try finding an older edition of a lower level course that you can buy off amazon for a few bucks that explain the applications. that may give you a more intuitive feel for what it is that you're doing. also, never forget youtube, which has a wealth of videos explaining concepts. I'm not sure what he has on linear, but patrickjmt makes great videos that explain concepts very well.

 

[zif.]

I think it's definitely the first math course where you actually have to sit down, read the textbook, memorize theorems, and actually learn the terminology.

What I found really helped me was making a glossary.

Not just reading the glossary in the text, but actually typing up the definitions of each new term at the end of the section or chapter.

 

[cdotter]

when you say "pure mathematics," what do you mean? do you mean proof based, doing things in the general sense, or just learning the lower level stuff, just with no real applications? linear alg was one of my favorite classes, on both levels. but it may have helped that the lower level class i took had a pretty easy book that helped explain things, like what this stuff is actually used for. my teacher loved pointing out that computer games use this stuff a lot, any time a 3d picture is shown, involving things rotating, or in a first person-shooter, for example, everything is stretched out depending on what angle you're looking at, which comes directly from this stuff.
try finding an older edition of a lower level course that you can buy off amazon for a few bucks that explain the applications. that may give you a more intuitive feel for what it is that you're doing. also, never forget youtube, which has a wealth of videos explaining concepts. I'm not sure what he has on linear, but patrickjmt makes great videos that explain concepts very well.

Pure, as in both my professor and my textbook mentioned none of the stuff I bolded in your post. My textbook can be found here http://linear.ups.edu/jsmath/latest/fcla-jsmath-latest.html

 

[stringy]

It appears the author has concerns about students becoming attached to their intution from two and three dimensions. He wants the students to be as comfortable in high dimensions as they are in low ones. Linear algebra is very general, which is where its power comes from. He wants to demonstrate that.

I can say, however, that I never felt hindered by my attachment to three dimensions. Whenever I tutor a student in linear algebra, I'm constantly throwing my arms around in the air pointing wildly to try and help them visualize what span, independence, etc. mean. It helps, I think.

Abstraction and generality should come later, when you can appreciate it and have a use for it. Having sat through an introductory linear algebra course (and being bored out of my mind), it was a revelation being introduced to operator theory and Hilbert space theory. That's when the theory "clicked" for me, and linear algebra suddenly became the most gorgeous branch of math.

I second the other recommendations for a different textbook. One with a heavier emphasis on the geometry. Are you a math major or science major?

 

[SpaceDomain]

I did not understand much of linear algebra the first time I took it but I have since taken a few classes that uses concepts like eigenvalues and linear transformations and these concepts have more meaning when you learn how they are applied.