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currently I am studying Vector spaces and sub spaces. I enjoyed working with matrices and using the Gaussian-Jordon elimination and I also had no problems with cofactor expansion and determinants in general. But for some reason I lost track when it came to vectors. I understand the geometrical representation of $\mathbb R^2$ and $\mathbb R^3$ and how to solve for angles and areas of a parallelogram.

I have a hard time thinking abstractly and I think that this is currently the problem why I don't grasp vector spaces. Do you have any advice on studying this material. I know I am not brilliant in math yet, but I want to study it and take more advanced topics because I see the beauty in math and how it applies to the real world. When I work through the proofs I am unable to see the turning point or the "a-ha" effect. The proofs are not in numbers so I can't even check my results if I am doing it right. Is there actually a method to train abstract thinking? I really appreciate any advice on this matter even though it is not the usually question asked here.

Thank you for your time reading this and your effort in possible answers.

-Daniel

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So I guess this isn't really a linear algebra question. There are many resources and books in techniques of mathematical proof, and many math departments at universities offer a course dedicated to teaching students the skill. I believe "How to Prove It" by Velleman is a well-regarded book on the topic. –  kigen Dec 12 '12 at 7:09
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The trick, especially when you're first starting out, is to really think about the material for hours. There is no short-cut. Just sit and ponder, play with it, play with examples, etc. Eventually, the nature of these abstract spaces will start to become familiar. –  providence Dec 12 '12 at 7:13
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3 Answers 3

Daniel, I understand what you mean. Sometimes it's difficult to understand abstract proofs, especially when the proofs only involve many variables/symbols/greek letters and not numbers.

What usually works for me is to get 1 or 2 examples (either from books, online, lectures, etc.) which illustrate how the theorem works. I feel that this "concreteness" helps me understand what is going on better, and then I go back and re-read the proof the of theorem. Usually easy, trivial examples are best.

Then, when I've understood the theorem and its proof well, I can try to apply it to more difficult examples.

So for example in Linear Algebra, when learning the Rank-Nullity Theorem, you could find a very simple example (maybe 2x2 matrices) from a textbook first to see it in practice. And perhaps try another yourself to see the Rank-Nullity Theorem in practice. Then afterwards go back and re-read the proof.

Hope that helps.

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One of the first things to do when learning about vector spaces is to see a lot of examples and work out why these examples are in fact vector spaces.

The first examples to look at should be the familiar ones (so you should try to convince yourself that the spaces $\mathbb{R}$, $\mathbb{R}^2$ and $\mathbb{R}^3$ satisfy those axioms you have been given for a vector space).

Next, one should try to look at some less familiar examples to get an idea of what sort of "other" things are vector spaces. Here are a few examples that may help to illustrate the ideas:

The set of polynomials of degree at most $2$ with the addition and scalar multiplication given by $(a_2x^2 + a_1x + a_0) + (b_2x^2 + b_1x + b_0) = (a_2+b_2)x^2 + (a_1+b_1)x + (a_0+b_0)$ and $\alpha(a_2x^2 + a_1x + a_0) = (\alpha a_2)x^2 + (\alpha a_1)x + (\alpha a_0)$.

The set of $n\times n$ matrices with the usual addition and scalar multiplication.

The positive reals with the addition defined as $a\oplus b = ab$ (usual multiplication in the reals) and scalar multiplication defined by $\alpha\otimes a = a^{\alpha}$ (usual exponentiation of real numbers).

The set of all functions from $\mathbb{R}$ to $\mathbb{R}$ with addition and scalar multiplication defined point-wise.

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The way I was taught to look at vector space and subspace is by looking at it with sets of arrows. Then you can break it down into sub arrows -- subspace -- and this is where bases, range, etc. comes into formation. Also, whenever you do the row-reduction technique, look at it as taking any set of vectors and finding how many unique arrows you need to represent all the arrows in your matrix which then will give you the dimension.

I was also taught that the arrows, in linear algebra, are not geometric vectors; they are only elements that behave like vectors at an abstract level. Also, one way that helped me get an intuitive understanding of vector space and subspace is by imagining it as a floor in a building. Each floor is a vector space -- whether it is in $\mathbb{R}$, $\mathbb{F}$, $\mathbb{C}$, etc. -- and each room inside that floor is a subspace and if you step out of that room you are stepping out of the subspace. It took me awhile to have an intuitive understanding, so you aren't alone. Hope this helped!

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