homeomorphic said: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 said: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 .
Nano-Passion said: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 said: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. .
chiro said: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 said: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 said: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.
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.
Linear algebra is a branch of mathematics that deals with the study of linear equations and their properties. It is crucial in science because it provides a powerful tool for solving complex problems and analyzing data sets. Many scientific fields, such as physics, engineering, and computer science, heavily rely on linear algebra for their mathematical models and calculations.
To improve your understanding of linear algebra, it is important to practice solving problems and working with matrices, vectors, and linear transformations. It is also helpful to study the underlying concepts and theories, such as vector spaces, eigenvalues, and determinants. Additionally, seeking out resources such as textbooks, online tutorials, and practice exercises can also aid in understanding.
Some common mistakes to avoid when working with linear algebra include forgetting to properly define and label variables, miscalculating matrix operations, and not understanding the properties and rules of vector spaces. It is also important to check for errors and simplify expressions as much as possible to avoid making mistakes.
Linear algebra can be applied to a wide range of real-world problems, such as data analysis, image processing, and optimization. It can also be used in fields like economics, genetics, and machine learning. By understanding the concepts and techniques of linear algebra, you can apply them to solve complex problems and make predictions based on data.
To effectively proof your work in linear algebra, it is important to check for errors and make sure all steps and calculations are clearly shown. It can also be helpful to double-check your answers by plugging them back into the original equations and verifying that they satisfy the given conditions. Additionally, seeking feedback from others or using online tools for proofing can also help to catch any mistakes.