Need help understanding Linear algebra proofs (and linear algebra in general)

[Sysman]

Hey all,

I am trying to get a head start on Linear Algebra before i start taking classes in a couple weeks. I am about to go into my second year undergraduate and all i have behind my belt is calculus (single varialbe, multivariable, and vector analysis (curl, divergence, etc)).

I am having a ton of trouble wrapping my head around stuff like vector spaces. Spans are especially difficult for me to understand because i can't visualize them. I understand what is said when i read the book, but it doesn't help me with go further because of the way it's presented.

I am using Axler's Linear Algebra Done Right to study from and here is an excerpt from chapter 2 about the let me i can't understand.

It is used for this proof

How can I adjust my way of thinking as to not get bogged down by the the strange new world of linear algebra? It has new notation, new ideas, and new things that i simply can't understand!

 

[chiro]

Hey Sysman and welcome to the forums.

The easiest way to think of it is as follows: a space will have some sort of basis. The basis will consist of so many linearly independent "vectors" and the right linear combination of said basis vectors can be used to describe any point in that space.

Usually we look at bases like the standard ones (i,j,k in 3D) but these are not the only ways (they don't even have to be orthogonal but there are reasons why we use orthogonal bases).

So the dimension of the space is equal to the minimum number of linearly independent vectors you need to describe the space.

Now you can get a subset of all vectors in the vector space and create a "basis" for this set (i.e. reduce it to a set of vectors where all elements of our subset can be written as a linear combination of some of the vectors in the subset).

You may end up finding that you have instances where you can construct a basis that has a less number of vectors than the basis for the whole space because a lot of the vectors are linearly dependent with the other ones.

In the above case, this is why you get the inequality as opposed to the equality.

To understand this as an example imagine you have (1,2,3,4) (2,4,6,8) (3,6,9,12): you know that all vectors in this space are linearly combinations of (1,2,3,4) so you only need one vector to describe the information in this set.

But of course in the whole space of R^4 you need four linearly independent vectors, but you happen to have chosen a special subspace where they are all linearly dependent.

 

[Muphrid]

I am having a ton of trouble wrapping my head around stuff like vector spaces. Spans are especially difficult for me to understand because i can't visualize them. I understand what is said when i read the book, but it doesn't help me with go further because of the way it's presented.

If you have only one vector, the span of that vector is a line through the origin. Every point on this line is a scalar multiple of the vector.

If you have two vectors, then as long as those two vectors aren't collinear, the span is a plane through the origin. Every point on that plane is a linear combination of the two vectors.

The span of a set of vectors defines a vector subspace--every vector that lies in the span is a linear combination of other vectors in the span.

 

[genericusrnme]

Is that the linear algebra done right that refuses to use determinants?
If so try swapping to Gilbert Strang's Introduction to Linear Algebra, it's a much nicer transition from no rigour to easy rigour.

 

[Stephen Tashi]

Spans are especially difficult for me to understand because i can't visualize them

It's nice to have a geometric picture of linear algebra concepts, but insisting on this can cripple your understanding! (and limit it to dimensions of 3 or less). Try to think about a variety of vector vectors spaces, not merely geometric spaces. Think about what the definitions and theorems say for other examples. The book probably gives examples such as spaces of polynomials, spaces of polynomials with coefficients in finite fields, etc. Linear algebra is not a subject that is exclusively about geometric space. Hence, not all of its concepts can be visualized.

 

[Sysman]

Is that the linear algebra done right that refuses to use determinants?
If so try swapping to Gilbert Strang's Introduction to Linear Algebra, it's a much nicer transition from no rigour to easy rigour.

Does Strang's text have an emphasis on proofs? I need learn how to do proofs for the linear algebra class I'm taking and I haven't had any formal proof-based class yet (this will be my first one).

 

[Vaedoris]

I have that (linear algebra done right) book too and I love it, but I have to admit that it is probably not suitable for some beginners in linear algebra.

On the other hand, the book by G. Strang is too wordy to my liking even though it covers many topics that one needs to learn on introductory level.

 

[mathwonk]

one problem is your book, not written for a first course.

you might try:

http://www.math.brown.edu/~treil/papers/LADW/book.pdf

and maybe also:

https://www.amazon.com/gp/product/0879011211/?tag=pfamazon01-20

 

[genericusrnme]

Does Strang's text have an emphasis on proofs? I need learn how to do proofs for the linear algebra class I'm taking and I haven't had any formal proof-based class yet (this will be my first one).

Yes, he does prove everything iirc, the proofs aren't complex or anything, they're all pretty much direct.

On the other hand, the book by G. Strang is too wordy to my liking even though it covers many topics that one needs to learn on introductory level.

I do agree that he's quite wordy but imo you have to start off being english wordy before you can become maths wordy. To me, jumping into a theorem - proof - corollary type book that has little to no exposition is like jumping straight into the Si Da Ming Zhu before you know how to ask for directions in Chinese.

 

[Mariogs379]

Hey Sysman,

I took linear algebra a couple years ago but have a pretty good handle on it. Wanna add me on Skype and I'll try to answer any questions you have? My handle is Claymangs07.


Mariogs

PS We used Knop's Linear Algebra: A First Course, and I thought it was great.

 

[homeomorphic]

It's nice to have a geometric picture of linear algebra concepts, but insisting on this can cripple your understanding! (and limit it to dimensions of 3 or less). Try to think about a variety of vector vectors spaces, not merely geometric spaces. Think about what the definitions and theorems say for other examples. The book probably gives examples such as spaces of polynomials, spaces of polynomials with coefficients in finite fields, etc. Linear algebra is not a subject that is exclusively about geometric space. Hence, not all of its concepts can be visualized.

I always think about it geometrically, whatever the content is. If it's 5th degree polynomials, I imagine it as a 5-dimensional space, etc. If it's over a different field than R, I still just pretend like it's a real vector space for some purposes, but I just keep in mind there are some differences. Never gets in the way of my understanding at all. Perhaps, I don't insist on every last thing being visual, but predominantly, that's the way I think about it, whether or not I am studying something that is explicitly "a geometric space", and it never hurts me to think of it that way, if I just use a little care.