Determinants and an upcoming midterm

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polymer
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Determinants and an upcoming midterm

Postby polymer » Thu Feb 25, 2010 2:49 am UTC

Hello! In linear algebra determinants were thrown at me last week, and now I'm 90% sure my next midterm is going to largely depend on them. My professor is pretty good, he's good about taking time to explain things that need to be explained. The Textbook is a real hit and miss though, in general it is not a very useful resource. I'm going to be spending some time on Wikipedia, but I was curious if there were any other online resources that people find useful for reviewing linear algebra.

To give an idea of where I'm at...

For those that may be familiar the text book, Its title and author is "Linear Algebra and its applications" by David C. Lay. And the midterm is going to be on chapters 2 to 3. Not including 2.6 and 2.5.


For those that aren't familiar with the text book.

This may be a little more exhaustive then necessary, but I figured it'd help me review if anything.

The things That have been covered include inverse matrices, their interpretation as elementary row operations, Linear transformations, how all of these things relate to the invertible matrix theorem, partitioned matrices(not hard), Linear transformations for manipulating vectors(like translating, rotating, scaling, shearing, etc.), subspaces including all of the vocabulary associated with it (Rank, Dim, Null space, column space, etc.), their relationship to basis vectors and the invertible matrix theorem, finding determinants, the effect of row operations on determinants, and finally Cramer's rule.

There are a couple proofs and theorems the book doesn't cover very well, the ones that come to mind mainly involve determinants.

A nice proof of why det(AB) = det(A)det(B) Would be sincerely appreciated. The book's proof brushes over it quickly, the concept is extremely useful for understanding the relationship between row operations and determinants, so a nice proof that can make clear why that works would be awesome.

Finally, the book mentioned that determinants return the area of parallelograms and the volume of parallelepipeds. It also gave some justifications illustrating how elementary row operations would affect the vectors of these shapes, and their relationship with the volume/area of the discussed parallelogram/parallelepiped. However I would very much prefer an explanation that develops to the definition of the determinant, beginning with the question of how one finds the area of these shapes.

The book basically told me the definition of the determinant, and illustrated all of its properties. I want to actually know what's going on with that recursive definition, why its constant, what inspired it, etc. I'm suspicious this is a simple concept explained poorly, but I have no idea. It almost looks like the determinant calculates the orthogonal components of all the vectors, but I don't know how the recursive definition implies that, or how I would derive the recursive definition given that goal.

Links and Resources would be nice, if you want to explain something that's okay to. Things that I'm particularly interested in at this point though, is what exactly is the determinant doing, how can I derive the definition of the determinant if I know what it's doing, and why the det(AB) = det(A)det(B).

Any help would be sincerely appreciated.

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t0rajir0u
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Re: Determinants and an upcoming midterm

Postby t0rajir0u » Thu Feb 25, 2010 4:06 am UTC

polymer wrote:A nice proof of why det(AB) = det(A)det(B) Would be sincerely appreciated.

If you like the row operation way of looking at things, show that this is true for elementary matrices and then write A and B as a product of elementary matrices. But the real content of this identity is geometric: it says that if A dilates volumes by a factor and B dilates volumes by another factor, then AB dilates by the product of those factors.

polymer wrote:Finally, the book mentioned that determinants return the area of parallelograms and the volume of parallelepipeds. It also gave some justifications illustrating how elementary row operations would affect the vectors of these shapes, and their relationship with the volume/area of the discussed parallelogram/parallelepiped. However I would very much prefer an explanation that develops to the definition of the determinant, beginning with the question of how one finds the area of these shapes.

Again, if you like the row operation way of looking at things, think about how this works for each row operation individually. For example, for adding a multiple of a row to another row, the picture you should have in mind is of a triangle whose base is on the x-axis and whose third vertex is allowed to vary in x-coordinate: the base and height stay the same, so the area stays the same, even though the triangle has been "skewed."

I know of a place online where these ideas are described in full generality, but it uses exterior algebra, which you probably don't need to be exposed to yet.

polymer wrote:The book basically told me the definition of the determinant, and illustrated all of its properties. I want to actually know what's going on with that recursive definition, why its constant, what inspired it, etc. I'm suspicious this is a simple concept explained poorly, but I have no idea. It almost looks like the determinant calculates the orthogonal components of all the vectors, but I don't know how the recursive definition implies that, or how I would derive the recursive definition given that goal.

Rest assured that this isn't a simple concept explained poorly; it's a really interesting concept made to look boring because the real explanation is hard.

In the first part of your question, I think you're talking about expansion by minors. Try to draw a diagram showing why the area of a parallelogram with vertices [imath](0, 0), (a, b), (c, d), (a+c, b+d)[/imath] is [imath]ad - bc[/imath] (i.e. the determinant of the matrix whose entries are the vectors). The idea is that [imath]ad[/imath] describes a larger area than the one you need and [imath]bc[/imath] is the difference. This idea generalizes to higher dimensions, but it's much harder to see.

If you want a really conceptual explanation as to what's going on, you probably want to think of the determinant as the unique alternating multilinear functional (up to a constant) on n vectors in n-space. Why should area / volume be alternating and multilinear? Well, multilinearity is just the picture I described above of the skewed triangle whose area stays the same; it means that if you add a multiple of a row to any other row, the determinant stays the same. The alternating property is the really interesting one; it means that the determinant switches sign when two rows are switched. The geometric meaning of this is that area / volume should be signed, i.e. it should have a positive and negative orientation. For example, a reflection changes orientations, so its determinant will be -1. There's a lot of interesting stuff here, most of which an introductory course in linear algebra will never get to.

If you're really interested in learning more about this point of view, you might want to try Sergei Winitzki's Linear Algebra via Exterior Products. But this book will take some patience and mathematical maturity to understand.

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Re: Determinants and an upcoming midterm

Postby njperrone » Thu Feb 25, 2010 4:11 am UTC

polymer wrote:Finally, the book mentioned that determinants return the area of parallelograms and the volume of parallelepipeds. It also gave some justifications illustrating how elementary row operations would affect the vectors of these shapes, and their relationship with the volume/area of the discussed parallelogram/parallelepiped. However I would very much prefer an explanation that develops to the definition of the determinant, beginning with the question of how one finds the area of these shapes.


This is done through the cross product. The cross product, denoted AxB where A and B are (sadly) 3 dimensional vectors. And the vectors A and B corresponds with the sides of the Parallelogram. What the cross product does is takes any two vectors and outputs another vector that is orthogonal to A and B. It uses a form of a determinate to find the vector AxB, and when you find the norm of AxB it gives you a scalar which is the area of the parallelogram. The volume of the parallelepiped is given by the triple-scalar product which has another vector C, which represents a leg of the piped, who's dot product with AxB gives the volume of the piped. http://en.wikipedia.org/wiki/Cross_product This wikipedia page should help with what i said and the use of determinates for the cross product. Sorry for not taking the time to put the determinate here, but I dont know how.

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Re: Determinants and an upcoming midterm

Postby t0rajir0u » Thu Feb 25, 2010 9:01 am UTC

I wouldn't invoke the cross product if I were you; among other things, it doesn't generalize (in the way you think it does) to more than three dimensions (except seven), whereas the determinant has the same meaning in all dimensions. If you read Winitzki you'll learn that the cross product and the determinant are both special cases of a more general construction.
Last edited by t0rajir0u on Thu Feb 25, 2010 7:10 pm UTC, edited 1 time in total.

polymer
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Re: Determinants and an upcoming midterm

Postby polymer » Thu Feb 25, 2010 6:34 pm UTC

Thanks for the interpretations, I'll take the qualitative understanding of volumes more seriously if the generalization is really that complicated. Thanks for posting a paper discussing the generalization as well, I'm not mature enough mathematically from what I can see at first glance, but it's always fun to give it a shot! I just see determinants everywhere, and knowing a full consistent way to relate all of them would be nice, rather then finding how the systems of equations set themselves up in each case(like the cross product as a special case of the determinant). Thanks again for the help! I'm sure this viewpoints will prove to be useful :)

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Re: Determinants and an upcoming midterm

Postby Yakk » Thu Feb 25, 2010 8:12 pm UTC

So, start with elementary matrices, and what the determinate does to them.

Next, confirm that the product of the determinate of elementary matrices is the determinate of the product of elementary matrices. This is important! (and somewhat tricky)

Now, imagine starting with an n-cube. Imagine transforming it by each elementary matrix. Convince yourself that the (oriented) volume of this cube is changed by a factor equal to the determinate of the elementary matrix.

Next, you have to convince yourself about what the volume of an n-dimensional object is, and that a linear transformation on it will change its "volume" in the same way that it changes the volume of the unit n-cube. (this requires defining what the n-volume of an n-dimensional object is, which can be hard at your level of expertise)

Now construct the inverse matrix of a product of elementary row matrices. Do you see how to do it? (arrange the inverses of each matrix in the original product, backwards). Note that the inverse of each elementary row matrix is an elementary row matrix, and the determinate is the inverse of the original elementary matrix! (with the notable exception of an elementary row matrix with a determinate of 0 -- the "erase a row" elementary row matrix -- which has no inverse)

Now you have a proof that describes how the determinate acts on products of elementary row matrices. The next step would be to prove that all matrices can be produced as a product of elementary row matrices.

From that, det(AB) = det(A)*det(B) falls out naturally, because A = product( E_A_0 ... E_A_n ), and B = product( E_B_0 ... E_B_k ) where the E_A and E_B's are elementary row matrices multiplied together.
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Cleverbeans
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Re: Determinants and an upcoming midterm

Postby Cleverbeans » Thu Feb 25, 2010 8:22 pm UTC

Gilbert Strang does an excellent lecture on the determinant function at the MIT OpenCourseWare site.
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Re: Determinants and an upcoming midterm

Postby njperrone » Fri Feb 26, 2010 1:57 am UTC

t0rajir0u wrote:I wouldn't invoke the cross product if I were you; among other things, it doesn't generalize (in the way you think it does) to more than three dimensions (except seven), whereas the determinant has the same meaning in all dimensions. If you read Winitzki you'll learn that the cross product and the determinant are both special cases of a more general construction.


I am very uninformed about the cross product other than its definition in 3 dimensions. Unfortunately the best response I could get from any of my math professors was "well because" when I asked them about it. And I go to a community college and lack faith in my current multivariable calculus professor. So, I was just saying what I know. Do you have any links to Winitzki?

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Re: Determinants and an upcoming midterm

Postby t0rajir0u » Fri Feb 26, 2010 4:24 am UTC

The link's at the end of my first post above. The cross product, like the determinant, is one of those things that can't be properly explained in an introductory linear algebra or multivariable class.

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Re: Determinants and an upcoming midterm

Postby njperrone » Fri Feb 26, 2010 6:18 am UTC

Oh, that link. I seem to have not seen it. Thanks!

polymer
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Re: Determinants and an upcoming midterm

Postby polymer » Sat Feb 27, 2010 1:56 am UTC

Finished the midterm, feel like I scored well.Thanks a bunch for the help, really appreciate it!

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Re: Determinants and an upcoming midterm

Postby polymer » Tue Mar 02, 2010 3:53 am UTC

Got a 94%! Thanks again for the help!


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