Proofing Linear Algebra: Tips, Advice and Pointers

[Nano-Passion]

I have done some "proofs" before in calculus. At this moment I am required to write proofs for linear algebra and I find them highly unintuitive and confusing -- I often don't know where to begin or what to do.

Can you guys leave some pointers, tips, advice, etc. for how to prove things, particularly for linear algebra.

 

[homeomorphic]

It's good to get practice with proofs in a pretty easy setting (like naive set theory), first, where there's no real difficulty in coming up with ideas, so that you can focus on getting used to logical arguments, things like proof by contradiction, working backwards from your goal and so on.

Another aspect of doing proofs is that you have to understand the concepts. That's where the ideas come from. Span, basis, dimension, linear independence, subspaces, linear transformations. Those are the main ones. It's a good idea to start by thinking about the simplest cases (here, that pretty much means dimensions 1-3, where you can visualize it all).

The intuition behind these concepts might not be spelled out very well in books. Usually, linear independence is defined by some equation that says a set of vectors is linearly independent if the coefficients are zero whenever the whole thing is zero. That's something you have to know, but it doesn't give you a good intuitive feel for the concept. To get a good intuitive feel, you have to see it sort of visually and think of low-dimensional examples. It's not always obvious how to do that. Two vectors are linearly independent if they span a plane, instead of a line. Another way to think of linear independence is that if a bunch of vectors are linearly dependent, you can throw some out without changing the span. But if they are linearly independent, throwing any vectors out will make them span a smaller space. You have to play these kinds of games in order to really understand linear algebra.

As for where to begin, you want to figure out what it is you are trying to prove and work backwards from there. Then, once you have figured that out, you can look at what you are given and see what you can do with it to work towards that goal.

 

[Nano-Passion]

It's good to get practice with proofs in a pretty easy setting (like naive set theory), first, where there's no real difficulty in coming up with ideas, so that you can focus on getting used to logical arguments, things like proof by contradiction, working backwards from your goal and so on.

Another aspect of doing proofs is that you have to understand the concepts. That's where the ideas come from. Span, basis, dimension, linear independence, subspaces, linear transformations. Those are the main ones. It's a good idea to start by thinking about the simplest cases (here, that pretty much means dimensions 1-3, where you can visualize it all).

The intuition behind these concepts might not be spelled out very well in books. Usually, linear independence is defined by some equation that says a set of vectors is linearly independent if the coefficients are zero whenever the whole thing is zero. That's something you have to know, but it doesn't give you a good intuitive feel for the concept. To get a good intuitive feel, you have to see it sort of visually and think of low-dimensional examples. It's not always obvious how to do that. Two vectors are linearly independent if they span a plane, instead of a line. Another way to think of linear independence is that if a bunch of vectors are linearly dependent, you can throw some out without changing the span. But if they are linearly independent, throwing any vectors out will make them span a smaller space. You have to play these kinds of games in order to really understand linear algebra.

As for where to begin, you want to figure out what it is you are trying to prove and work backwards from there. Then, once you have figured that out, you can look at what you are given and see what you can do with it to work towards that goal.

Thank you Homeomorphic.

Your right, I need to spend more time visualizing things. I almost didn't give an effort for linear algebra, although I always do it for other subjects. It might have be due to the nature of linear algebra, it generally appears to be a bit more abstract .

And I wanted to work on "how to prove it: a structured approach" -- but my week has been rather busy. I should have started it in the summer.

 

[chiro]

One suggestion I have is if you are going through a lot of definitions that sound almost made up, then take that definition and create a visual definition somewhere beside it and put all of these together in one spot for later reference.

This can help when you see that you have to prove some theorem with a lot of funny words that otherwise might look completely non-sensical (but visually, make a lot of sense).

Even if the result is for a general vector space, drawing a diagram with arrows still can clarify things even for the general case.

 

[SophusLies]

I almost didn't give an effort for linear algebra, although I always do it for other subjects. It might have be due to the nature of linear algebra, it generally appears to be a bit more abstract .

I hope you don't plan on taking quantum mechanics, because all that abstract linear algebra will come back to haunt you if you didn't take it seriously the first time.

On proofs in general, the only way to really learn them is doing them. You don't go to the track and watch others run to train to run do you? Why would it be any different doing math?

 

[Dembadon]

I have done some "proofs" before in calculus. At this moment I am required to write proofs for linear algebra and I find them highly unintuitive and confusing -- I often don't know where to begin or what to do.

Can you guys leave some pointers, tips, advice, etc. for how to prove things, particularly for linear algebra.

Linear algebra is a bit more friendly to visual learners than other topics in mathematics, so as others have suggested, visualizing can certainly help you develop your arguments.

However, there are going to be some things that are very difficult to visualize (n-dimensional structures). With respect to writing proofs, it is quite common for the beginner to not know where to start.

I think it's very important to internalize the relevant definitions and theorems. Take a look at what you're being asked to prove and see if there is a theorem or definition that is relevant in any way. You might have to go back a few sections. Make sure you know what every single word/concept in the problem means, and lookup things that aren't clear to you, even if only in the slightest manner.

 

[clope023]

Two vectors are linearly independent if they span a plane, instead of a line. Another way to think of linear independence is that if a bunch of vectors are linearly dependent, you can throw some out without changing the span. But if they are linearly independent, throwing any vectors out will make them span a smaller space. .

I always thought of it like vectors being independent means they are not dependent on any of the other vectors in the space for its length, ie v1 =/= cv2 and expanding like that; but I suppose you said the same thing here.

 

[Nano-Passion]

One suggestion I have is if you are going through a lot of definitions that sound almost made up, then take that definition and create a visual definition somewhere beside it and put all of these together in one spot for later reference.

This can help when you see that you have to prove some theorem with a lot of funny words that otherwise might look completely non-sensical (but visually, make a lot of sense).

Even if the result is for a general vector space, drawing a diagram with arrows still can clarify things even for the general case.

Awesome advice, thanks!

I hope you don't plan on taking quantum mechanics, because all that abstract linear algebra will come back to haunt you if you didn't take it seriously the first time.

On proofs in general, the only way to really learn them is doing them. You don't go to the track and watch others run to train to run do you? Why would it be any different doing math?

Your absolutely right. I am actually very big on intuitively thinking about things and not getting lost in the mathematics, just not sure why I gave up on linear algebra. But If I want to do good on quantum mechanics, I will have to master linear algebra in the intuitive sense.

Linear algebra is a bit more friendly to visual learners than other topics in mathematics, so as others have suggested, visualizing can certainly help you develop your arguments.

However, there are going to be some things that are very difficult to visualize (n-dimensional structures). With respect to writing proofs, it is quite common for the beginner to not know where to start.

I think it's very important to internalize the relevant definitions and theorems. Take a look at what you're being asked to prove and see if there is a theorem or definition that is relevant in any way. You might have to go back a few sections. Make sure you know what every single word/concept in the problem means, and lookup things that aren't clear to you, even if only in the slightest manner.

Thanks, your advice will prove very helpful. All of the advice given to me so far is very solid-- thanks everyone.

 

[homeomorphic]

I always thought of it like vectors being independent means they are not dependent on any of the other vectors in the space for its length, ie v1 =/= cv2 and expanding like that; but I suppose you said the same thing here.

Not sure that makes any sense. If one vector is independent of a bunch of other vectors, it means it is not "lying flat" in the subspace they span. Maybe what you are trying to say is that if a vector is dependent on others, then you can hit it with some linear combo of the other vectors. That's more or less the same thing as what I said earlier.