What Prerequisites Should I Review Before Learning Linear Algebra?

[Saladsamurai]

So I just bought his ook called "Matrices and Linear Algebra" by Hans Schneider and G.P. Barker in an attempt to teach myself this subject as I will probably need to learn it at some point anyway. I am not very familiar with the topic (in any sense) and I was wondering what if any topics up through Calculus 1 I should review (if any of these would even help).

Any thoughts are appreciated. For now I'll just wiki the subject and try to get a general feel of what it even entails...I don't even know what the term Linear Algebra means beyond a linear equation (of the most basic form).

Casey

 

[Pseudo Statistic]

You don't really need any calculus to learn linear algebra-- unless you want to go through applications.
Like, diagonalization can be used to solve a system of linear, homogeneous ODEs-- so, if anything, the most you need is basic knowledge of separation of variables. (i.e. $$\frac{dy}{dt}=ky$$)
However, from skimming through the book, it looks like it covers the background you need for the application I mentioned-- so I guess basic knowledge of integration would do.

 

[Saladsamurai]

You don't really need any calculus to learn linear algebra-- unless you want to go through applications.
Like, diagonalization can be used to solve a system of linear, homogeneous ODEs-- so, if anything, the most you need is basic knowledge of separation of variables. (i.e. $$\frac{dy}{dt}=ky$$)
However, from skimming through the book, it looks like it covers the background you need for the application I mentioned-- so I guess basic knowledge of integration would do.

I see. I am not familiar with ant of those terms, nor does the expression $$\frac{dy}{dt}=ky$$ have any meaning to me; hopefully I will be able to get that out of the reading.

Casey

 

[Pseudo Statistic]

Don't worry about it-- neither was I. ;)
If you're not familiar with that differential equation, you might need to learn a bit of calculus while studying linear algebra-- but like I said, you don't need it to understand the material, it's mainly a convenience should you ever feel like applying all of the underlying material.
As for what the terms mean, again, don't worry about it... it'll become clearer as you go along.

 

[Zurtex]

I've always found linear algebra a little dry and unexciting, but certainly extremely useful in many areas of mathematics. If you stick with it, you might want to check out Abstract Algebra at some point, group theory is generally quite enjoyable to most maths geeks because it starts off so simple and yet has such far reaching consequences

I don't know the American education system, but once you can, you should enjoy showing that the differential operator is a linear operator.

 

[Saladsamurai]

The main problem I have with learning a new topic is all of the new definitions.

For example, I am still not sure what Linear Algebra even is! I looked it up on Wiki and it seems that in the definition alone there are new terms at every turn. This led to a chain of look-ups and at the end of it all, I was more confused than I was to start!

Anyone have a good "lay" definition of Linear Algebra for me?

~Casey

 

[CRGreathouse]

Anyone have a good "lay" definition of Linear Algebra for me?

In a nutshell, it's solving systems of equations (usually in real numbers or matrices), and the special tools used toward that end.

 

[Saladsamurai]

In a nutshell, it's solving systems of equations (usually in real numbers or matrices), and the special tools used toward that end.

Nice CRGreathouse. That actually helps a lot.

~Casey

 

[radou]

In a nutshell, it's solving systems of equations (usually in real numbers or matrices), and the special tools used toward that end.

Many problems in linear algebra lead to solving a system of linear equations, although this very "problem" is almost trivial from a pure theoretical standpoint.

 

[Saladsamurai]

Many problems in linear algebra lead to solving a system of linear equations, although this very "problem" is almost trivial from a pure theoretical standpoint.

I am not sure that I quite follow you radou? Though I am sure it is due to my being new to the topic.

 

[matt grime]

What radou is saying is that the description you were given, of solving systems of linear equations, is not a very good one. That is an elementary topic taught in school. Linear algebra is the study of vector spaces and the maps between vector spaces. That is 'all' it is, in the sense of 'the' definition. If you know what a vector space is and a matrix, then you know the basics.

Solving linear equations is finding, if it exists, a point in the intersection of subspaces. It is merely algorithmic, tedious, and dull.

What are vector spaces? Well, let's not get into definitions, but let's content ourselves with saying that an understanding of them is a prerequisite for most any mathematics and physics - we use them as objects of interest in their own right, or use them in modelling physical situations.

 

[mathwonk]

it may not be much use , but there is a short (15 page) free linear algebra book on my webpage. a better one is the free one by ruslan sharipov.linear algebra and linear geometry are useful because they are easy, and yet can be used to give approximate answers to non linear and hence hard, problems.

differential calculus is the art of approximating non linear problems by linear ones. hence one does not need calculus to learn linear algebra, but one does need it to apply linear algebra most deeply, e.g. to differential equations.

 

[Saladsamurai]

Thank you Matt Grime and Mathwonk. I will check out that book(s) when I get out of work.

I literally just picked up a book a couple of days ago, so I have not come to the part where I discover that Linear Algebra isn't really about solving equations (at least in the way I thought).

Vector spaces and subspaces have no meaning to me...yet
I mean, I know what vectors are from physics 1, but that is the extent of it.

But, be sure that I look foward to learning this. I am going to look up those terms now. (Then I will probably have to search around PF for a translation!:/ )

Casey

 

[mathwonk]

the idea is to understand "linearity". i.e. operations that preserve addition and scalar multiplication, like differentiation does.

these are called linear operators, and the spaces they act on are called vector spaces.

so the space of infinitely differentiable functions on an interval is a vector space, and differentiation is a linear operator on it. that's basically what we want to understand.

 

[Ruslan_Sharipov]

My geocoities.com/r-sharipov site is now off. Use the following sites instead:
http://ruslan-sharipov.ucoz.com"
http://freetextbooks.narod.ru"