Easy definition of linear algebra

[PainterGuy]

hello fine people,

for a beginner which of these definitions of linear algebra is more correct your view:

1:- the part of algebra that deals with the theory of linear equations and linear transformation

2:- Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions which input one vector and output another, according to certain rules. ...

3:- in which the specific properties of vector spaces are studied (including matrices);

the definations above are from google. for some reason i like the number 1. please tell me your view.

cheers

 

[HallsofIvy]

I would prefer number 1 also. It does not make sense to talk about "vectors" and "vector spaces" until you have defined them. And you define them in Linear Algebra so it make no sense to define Linear Algebra itself as the study of "vectors" and "vector spaces".

 

[Fredrik]

In that case, it wouldn't make sense to mention linear transformations either, since you have to define "vector space" before you define "linear transformation". So I disagree with the suggestion that the definition shouldn't mention vector spaces. I don't have a definition of "linear algebra" that I think is definitely the best one, but what I'm thinking right now is that linear algebra is the part of the mathematics of linear functions between vector spaces that doesn't involve limits. (If it involves limits, it falls under functional analysis instead).

A couple of months ago, I would have said that the vector spaces need to be finite-dimensional, but consider e.g. the theorem that says that the smallest subspace that contains a given subset S, is the set of all linear combinations of members of S. This applies to infinite-dimensional vector spaces as well, and I think it would be pretty weird to say that it's not linear algebra. So I have changed my mind about where to draw the line between linear algebra and functional analysis, from "finite-dimensional vs. not necessarily finite-dimensional" to "doesn't involve limits vs. involves limits". But I might change my mind again before this thread is over.

 

[micromass]

I think all three of the definitions are quite interesting, as they present linear algebra from different point-of-views.

The first definition is the point-of-view of an applied mathematician. They won't be all that interested in vector space and all that, but they will be interested in it's applications i.e. solving linear equations and finding suitable linear transformations.

The second definition seems to me the point-of-view of the category theorist. A category theorist will see a mathematical theory as a collection of objects (the vector spaces) together with it's structure-preserving functions (the linear maps). The entire theory is in this viewpoint.

The third definition (my favorite) is the point-of-view of abstract algebra. They will care about an algebraic structure and will try to say as much as possible about that structure. They won't really care about applications, but more in the structure of a vector space itself.

All three definitions have their advantages and disadvantages. So in a good linear algebra course, all three definitions must be present...

 

[PainterGuy]

hello,

okay most of you discussion is over my head! someone once told me matrices initially were made to deal with linear equations. after that they were generalized for other applications as is case with many other things in math.

in school we once used 4x4 matrices for transformations. is linear algebra about those kind of transformation but on complex scale. please lead me on these things.

cheers

 

[micromass]

Well, linear algebra is about a lot of things. In some ways, it's a culmination of several fields, including algebra and geometry. But let's keep it basic.

The first instances of linear algebra that people usually meet are when dealing with systems of equations. To solve these more efficiently, people use the notation of matrices. Linear algebra then studies these matrices and defines several properties on them (like addition, multiplication, determinants). Thusfar, people have only studies matrices to solve equations. But geometry provides another great tool for matrices. Matrices can be seen as functions. For example, the matrix

$$\left(\begin{array}{cc}{1 & 2\\ 3 & 4\end{array}\right)$$

can be seen as the function

$$f:\mathbb{R}^2\rightarrow \mathbb{R}^2:(x,y)\rightarrow \left(\begin{array}{cc}{1 & 2\\ 3 & 4\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$$

So, we could ask the question: "what are the maps $$f:\mathbb{R}^2\rightarrow\mathbb{R}^2$$ that can be descirbed by matrices?" These are exactly the linear maps. The mathematicians and the physicist then generalized this concept to vector spaces and linear maps. So actually, linear algebra is simply the study of linear maps (like above) between vector spaces.

Of course, there is a lot missing in this story. For example, I have not mentioned physics which provided the framework for vector spaces. But as a mathematician, I don't feel qualified to talk about physics...

 

[PainterGuy]

Well, linear algebra is about a lot of things. In some ways, it's a culmination of several fields, including algebra and geometry. But let's keep it basic.

The first instances of linear algebra that people usually meet are when dealing with systems of equations. To solve these more efficiently, people use the notation of matrices. Linear algebra then studies these matrices and defines several properties on them (like addition, multiplication, determinants). Thusfar, people have only studies matrices to solve equations. But geometry provides another great tool for matrices. Matrices can be seen as functions. For example, the matrix

$$\left(\begin{array}{cc}{1 & 2\\ 3 & 4\end{array}\right)$$

can be seen as the function

$$f:\mathbb{R}^2\rightarrow \mathbb{R}^2:(x,y)\rightarrow \left(\begin{array}{cc}{1 & 2\\ 3 & 4\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$$

So, we could ask the question: "what are the maps $$f:\mathbb{R}^2\rightarrow\mathbb{R}^2$$ that can be descirbed by matrices?" These are exactly the linear maps. The mathematicians and the physicist then generalized this concept to vector spaces and linear maps. So actually, linear algebra is simply the study of linear maps (like above) between vector spaces.

Of course, there is a lot missing in this story. For example, I have not mentioned physics which provided the framework for vector spaces. But as a mathematician, I don't feel qualified to talk about physics...

a lot of thanks micromass. most of your post was easy to understand if not all.

R is set of real numbers, what is R^2. if this is some complex thing then just ignore it which i could be not able to understand.

i have heard this word a lot of time "vector space", what is this? is there some definition or explanation?

much obliged for all all the support. you all people are very helpful.

cheers

 

[micromass]

$$\mathbb{R}^2$$ is just the set of all couples of real numbers, i.e.

$$\mathbb{R}^2=\{(x,y)~\vert~x,y\in \mathbb{R}\}$$

Alternatively, $$\mathbb{R}^2$$ is the plane. (while $$\mathbb{R}$$ is the line).

A vector space is, informally, the most general space on which you can define linear transformation. I think wiki has a very good explanation on what a vector space is: http://en.wikipedia.org/wiki/Vector_space

 

[AlephZero]

okay most of you discussion is over my head!

Don't worry about that. When you started learing about numbers, you didn't start with a definition of what a number "is", you started by learning how to do some useful things with numbers, like how to count, add, and multiply.

The same goes for most parts of math. You won't really "understand what linear algebra is" or how to "define" it, until you know how to use it.

And one important thing about linear algebra (as the previous posts showed) is you can use it in whole range of (at first sight unrelated) applications in pure and applied math.

 

[PainterGuy]

$$\mathbb{R}^2$$ is just the set of all couples of real numbers, i.e.

$$\mathbb{R}^2=\{(x,y)~\vert~x,y\in \mathbb{R}\}$$

Alternatively, $$\mathbb{R}^2$$ is the plane. (while $$\mathbb{R}$$ is the line).

A vector space is, informally, the most general space on which you can define linear transformation. I think wiki has a very good explanation on what a vector space is: http://en.wikipedia.org/wiki/Vector_space

micromass, one time more thanks. wiki has an article on vector space. it will be good from your view. it's long full of english and difficult on my side. I'm only after some basic concept of vector space. if this is possible, please let me have some understanding. does adding two vectors v and w produce vector space?

AlephZero, what you say is not wrong. but it will be okay to know some basic applications and details. all people learn in totally different ways.

 

[bonker]

linear algebra = degeneracy.

 

[micromass]

micromass, one time more thanks. wiki has an article on vector space. it will be good from your view. it's long full of english and difficult on my side. I'm only after some basic concept of vector space. if this is possible, please let me have some understanding. does adding two vectors v and w produce vector space?

Well, it is my understanding that the concept of vector space can only be understood by seeing many examples. But in general, a vector space is simply a set on which we define an addition and a scalar multiplication, which behave nicely.
For example, an arbitrary element of $$\mathbb{R}^2$$ is (x,y). We can add two such elements by (x,y)+(x',y')=(x+x',y+y'). We can also multiply this elements by scalars, thus for $$\alpha\in \mathbb{R}$$, we can define $$\alpha(x,y)=(\alpha x,\alpha y)$$. Thus $$\mathbb{R}^2$$ forms a vector space.

Don't worry if you don't grasp this concept immediately, I didn't grasp it immediately to. It took me some years and many examples before I finally got it. You can't force understanding: you'll need to be acquainted with matrices and some geometry before you can understand what a vector space is...

 

[Bacle]

But part of the problem is that even the concept of linear map is not clear;
instead affine maps are referred to as being linear, and even non-affine ones, like:

L(x₁,x₂,...x_n )= a₁x¹+...+a_nxⁿ=b,

for which we cannot even test for L(x+y)=L(x)+L(y), nor for L(cx)=cL(x),

but at best, we can test for linearity in each variable separately.

The best definition of linearity of a map is that of a map in which solutions

(to Ax=0) are preserved under linear combinations, i.e., if Ax1=0 and Ax2=0

then A(cx1+dx2)=0 , and (maybe kind of contrived) in the non-homogeneous case

, if Ax1=b and Ax2=0, then A(cx1+dx2)=cAx1+dAx2 , but this last expression for

the non-homogeneous case is kind of contrived

 

[Robert1986]

I think that in the most abstract (and thus general) form, the third would probably be closest. Then, we can explain that one application of a particular type of vector space is in solving linear equations.

As someone has pointed out, these are really just isomorphic definitions.

For example, I took a class called linear algebra. I go to a major Engineering college and this class was focused almost entirely on using matrices to solve systems of equations, so the first definition would work. Our textbook introduced matrices operations on matricies as a way of solving systems of equations. Then, we started to learn about vector spaces, but we really just dealt with R^n.


Then, I took a class that was called "Abstract Vector Spaces" (or something like that.) The textbook we used was Apostol's Linear Algebra. The point of this course is to a)give a rigorous treatment of linear algebra b)teach math students how to write proofs. This book starts out by giving an axiomatic definition of a vector space. Then we study linear transformations of vector spaces. Then we study how matrices can be used to represent linear transformations.


In Algebra II, we use Herstein's Topics In Algebra. There, we look at vector spaces in terms of fields and modules (BTW, it went much easier than the first two times around.) I see Dummit and Foote does it the same way, too.


So, it seems that, as someone has said, there are many ways to define linear algebra, but I prefer the most general method. While I'm at it, I will share this joke:

What is the physicist's definition of a vector space?
A set V such that if v is in V, then v has a little arrow drawn over it.

 

[micromass]

I think that in the most abstract (and thus general) form, the third would probably be closest. Then, we can explain that one application of a particular type of vector space is in solving linear equations.

As someone has pointed out, these are really just isomorphic definitions.

For example, I took a class called linear algebra. I go to a major Engineering college and this class was focused almost entirely on using matrices to solve systems of equations, so the first definition would work. Our textbook introduced matrices operations on matricies as a way of solving systems of equations. Then, we started to learn about vector spaces, but we really just dealt with R^n.


Then, I took a class that was called "Abstract Vector Spaces" (or something like that.) The textbook we used was Apostol's Linear Algebra. The point of this course is to a)give a rigorous treatment of linear algebra b)teach math students how to write proofs. This book starts out by giving an axiomatic definition of a vector space. Then we study linear transformations of vector spaces. Then we study how matrices can be used to represent linear transformations.


In Algebra II, we use Herstein's Topics In Algebra. There, we look at vector spaces in terms of fields and modules (BTW, it went much easier than the first two times around.) I see Dummit and Foote does it the same way, too.


So, it seems that, as someone has said, there are many ways to define linear algebra, but I prefer the most general method. While I'm at it, I will share this joke:

What is the physicist's definition of a vector space?
A set V such that if v is in V, then v has a little arrow drawn over it.

That's a really great post, robert! I think you've been very lucky because you've seen linear algebra from three sides: from the application side, the axiomatic side and as a special case of modules. To really "get" linear algebra, you have to see it from these three sides. And most courses don't really see it from three sides. For example, I have seldom seen the application side of linear algebra, and I really feel that I'm missing something there.

Thesame is true for any math course: you need to be acquainted with both the theory and the applications. I think it is very unfortunate that engineering courses only focus on applications, while mathematicians only focus on the theory. I understand why they do this (there isn't enough time to cover everything), but I still think it's unfortunate...

 

[Studiot]

Can't say I'm knocked out by any of these definitions specifically for a beginner.

1:- the part of algebra that deals with the theory of linear equations and linear transformation

Well I started with Nering when I was a beginner and I don't think you will find many equations in Nering, although the book is pretty comprehensive.
Many aspects of linear alegebra are not to do with equations.

For example the Gram Schmidt process

Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions which input one vector and output another, according to certain rules. ...

Sometimes the input is a vector and the output a vector, but many times not.

For example Fourier analysis

in which the specific properties of vector spaces are studied (including matrices);

This one is a bit of a cop out and could mean anything.

"-Studies properties"

What properties?

All possible properties associated with vector spaces?
Some of the properties associated vector spaces? If so which ones and who gets to choose?
What about the ones that are omitted?

 

[Studiot]

Painterguy, sometimes definitions offer little or no explanation of their meaning. They are designed to be very short.

if you want want a simplified guide to what linear algebra is about, rather than a comment on your three versions, post a reply.