Free Algebra Notes: Linear Algebra, Group Theory, Riemann Roch

[mathwonk]

On the following site there are some free class notes, 1). on linear algebra, very brief, and 3). more extensive notes on group theory. There is also a set of notes 2). on the Riemann Roch theorem (these last use complex analysis of one and several variables, and algebraic and differential topology; but linear algebra plays a role in the guise of exact sequences of sheaf cohomology, i.e. what kempf calls "global linear algebra"),

(you are also welcome to the research preprints, mostly about principally polarized abelian varieties.)


http://www.math.uga.edu/~roy/


oh and here are some review questions for linear algebra:

Define a eigenvalue of a square matrix A.

Define an eigenvector of a square matrix A.

If a matrix A has a basis of eigenvectors, what does the new matrix for µA (multiplication by A) look like in that basis?

Explain the geometric meaning of the determinant of a square matrix A.

Give the formula for the determinant of a 2by 2 and a 3by 3 matrix A.

Explain how to find the determinant of any matrix by row reduction.

What is the determinant of a product of two matrices?

What is the determinant of a diagonal matrix?

What is the relation between the determinants of two “similar” matrices?

What is special about the determinant of an invertible matrix?

Define characteristic polynomial of a square matrix.

Tell how to recognize an orthogonally diagonalizable matrix.

Tell how to recognize any diagonalizable matrix.

Describe all length and orientation preserving linear maps of R^3, R^n.

Tell how to recognize the matrix of a reflection in a plane in R^3.

Tell how to recognize a matrix of an orthogonal projection in R^3.

If c is an eigenvalue of A, how do you find the eigenvectors for c?

If A is a diagonalizable matrix, how do you actually find a matrix P such that P^(-1)AP is diagonal?

 

[quasar987]

What would a good program to read *.ps be?


I would like to hear the answer of "Explain the geometric meaning of the determinant of a square matrix A.",

thx

 

[mathwonk]

well a square n by n matrix yields a map from R^n to R^n by multiplication. Applying this map to the standard unit vectors, carries the unit cube they span to another parallelepiped, or block. The determinant is the oriented volume of that block, i.e. it equals the volume of that block counted plus or minus acording to whether the map preserves or reverse orientation.

thus the determinant is ( plus or minus) the factor by which the map expands or contracts volume, and is zero if the map sends the block into a lower dimensional figure.

this is why it appears in the formula for change of variables in integration in several variables.


i do not know what *.ps means, unless you mean how to read my postscript notes. the way to deal with them is drag them into your printer icon and just print them out, sorry about that. i will slowly update everything to readable pdf files.

by the way there are 5 more parts of the abstract algebra notes that are not up yet, as you can deduce from the table of contents.

 

[jma2001]

mathwonk,

Thanks for those notes. Is that your home page?

 

[quasar987]

Mathwonk! I had no idea that was your personnal web page until I read "Linear algebra is about linear spaces or vector spaces, and linear maps between them." This is exactly how you started your little condensed course on linear albegra in an earlier thread on PF! I liked those posts of yours a lot. And you know what's funny? I was on the subway earlier and my mind was wandering and sudently I remembered that very sentence: "Linear algebra is about linear spaces and linear maps between them". Occult.

P.S. I wasn't imagining you like that.. you look somwhat...different then on your avatar.

 

[quasar987]

The geometrical interpretation of the determinant is fascinating mathwonk, but I can't get it to work on $\mathbb{R}^2$. :grumpy:

$$\left(\begin{array}{cc} 2 & 3 \\ 4 & 5 \end{array}\right)\left(\begin{array}{cc} 1 \\ 1\end{array}\right) = \left(\begin{array}{cc} 5 \\ 9\end{array}\right)$$

So here, the square matrix maps the square of unit length sides to a rectangle of sides 5 and 9, and hence of area 45. But the determinant is only

$$\left(\begin{array}{cc} 2 & 3 \\ 4 & 5 \end{array}\right) = 2\cdot 5 - 3 \cdot 4 = -2$$

 

[matt grime]

That isn't how the matirx transforms the unit square - why must the image be a recatngle? The unit square maps to a quadrilateral with vertices at

(0,0), (2,4) (3,5) and (5,9) which are the images of (0,0) (1,0) (0,1) and (1,1) resp. what is it's area? Allowing for any orientation flips too.

Hopefully later today I should have online some more pages of maths at

www.maths.bris.ac.uk/~maxmg[/URL]

now that I've brought my laptop into the office at last. An odd coincidence?

 

[matt grime]

Right, I've put it on the server, and some of the following may be interesting:

www.maths.bris.ac.uk/~maxmg/frontpage.html[/URL]
[PLAIN]www.maths.bris.ac.uk/~maxmg/maths[/URL]

still messy, will get renovated soon and pages will get finished, hopefully

 

[robphy]

.ps files are Postscript files and can be read by ghostview or gsview: http://www.cs.wisc.edu/~ghost/

Here's a followup question (whose answer I've been seeking for a while):

"Explain the geometric meaning of the Principal Invariants of a square matrix A"
The determinant and the trace are the nth and first principal invariants, respectively.. In particular, I'm interested in the second principal invariant: $$\displaystyle\frac{1}{2}\left( tr(A)^2 - tr(A^2) \right)$$

 

[mathwonk]

What appears here is an actual photograph, the one on the webpage is my avatar.

yeah, the posts here were the beginning of the brief linear algebra notes. They were inspired by something I saw here, maybe a reference Sharipov's excellent book, and how short it was. I was on xmas break with nothing to do for a couple of days and thought, "I can write a shorter book than that."

i was amazed how short one can make the treatment, even fairly self contained, just leaving out "tautological" and "straightforward" proofs (and row reduction and worked examples!)

I was also preparing (mentally) to teach linear algebra over spring semester. We only got through about half of those notes in the 15 week course! But we did a lot more with rotations, reflections and other length preserving maps.

 

[matt grime]

I'm not sure I know what the invariants are, but it looks like, since tr is the first and det the nth

that they are the coefficients of the characteristic polynomial which are related to the eigen values.

 

[mathwonk]

RE; matt's post #7, if you draw those vertices, you can compute the area of the parallelogram by combining (adding and subtracting) areas of several triangles and rectangles. You should get 2 as the area, and since the determinant is supposedly negative, you should note that the image of the vector (0,1) is clockwise from, rather than counterclockwise from, the image of (1,0).

 

[mathwonk]

I agree with matt on those coefficient - invariants. indeed i think those are essentially the only invariants of a linear map.

there are several situations where a certain invariant polynomial vcan be associated to an object and then one tries to interpret the coefficients which must be invariant too.

there is the hilbert polynomial of a projective variety, or more generally of a line bundle, and the former seem to be a certain invariant, the arithmetic genus, of the various linear slices of the embedded projective variety.

then there is the chern character of a vector bundle, where there are formal objects like eigenvalues called chern roots, and their invariant symmetric combinations may have soimehting to do with the various chern classes, or obstructions to the bundle having more and more independent sections. there is something about this in my riemann roch notes, but i was just learning the topic.

even for the identity map, where the eigenvalues are all 1, these invariants have somewhat questionably important geometric meaning, i.e. they are apparently the binomial coefficients of (x+y)^n.

I mean you are getting for the image of the unit cube, that the trace is 1/2^(n-1)? times the sum of the edge length of its image, and the determinant is the oriented volume. but the second invariant would not even be the sum of the areas of certain of the 2 diml faces, unless the standard unit vectors were an orthogonal eigenbasis. ?

 

[robphy]

Yes, I agree. I've got a short draft of an article written up on this, with results along the "sum of areas" notion... but I haven't been completely happy with it. So, it's been on my backburner for a while. I was hoping for something more. Thanks.

 

[quasar987]

That isn't how the matirx transforms the unit square - why must the image be a recatngle? The unit square maps to a quadrilateral with vertices at

(0,0), (2,4) (3,5) and (5,9) which are the images of (0,0) (1,0) (0,1) and (1,1) resp. what is it's area? Allowing for any orientation flips too.

Hopefully later today I should have online some more pages of maths at

www.maths.bris.ac.uk/~maxmg[/URL]

now that I've brought my laptop into the office at last. An odd coincidence?[/QUOTE]

Oh ok, mathwonk meant that in the sense that we let the matrix act on each corner of the square, instead of on the vector which "spans" the square in the sense I had understood it...

On this subject, here's a question I had always wondered about. Suppose we have a certain area in $\mathbb{R}^2$ and a linear map from $\mathbb{R}^2$ to $\mathbb{R}^2$. We want to find the corresponding area after applying the linear transformation on each point of the original area. To solve such a problem, the textbook I have simply finds the corresponding [I]contour[/I] and then conclude that the corresponding area lies inside that contour. But what assures us that this is so? That, for instance, the following is not possible? (see figure)

 

[Cincinnatus]

Hello,
I'm a long time reader and first time poster. I actually just took linear algebra in the fall...
Anyway, Of all those questions you posted the one I am least sure of was:

"Describe all length and orientation preserving linear maps of R^3, R^n."

I don't think my class covered this... or if it did I must have forgotten more than I thought.

 

[mathwonk]

in R^3 the answer is there is a fixed axis passing through the origin, and the map rotates around this axis.

In even dimensions the space decomposes into a sum of orthogonal; planes and there is a rotation in each plane. in odd dimensions there is one fixed axis and the orthogonal even dimensional space decompsoes as above.

not everyone does this but i thought rotations with fixed vectors, were more natural and familiar than general eigenvectors, so i focused on them.

 

[mathwonk]

quasar 987, that picture cannot occur for a linear isomorphism, since an isomorphism is also a continuous homeomorphism so cannot change the "connectivity" of the space, e.g. cannot change the fundamental group, or the number of pieces the complement breaks into.

the more general formula for area change munder possibly non linear maps uses the change of variables formula for several variable, integrals, and it says that the area of f(U), equals the integral over U of the function whose value at each point p of U is the determinant of the jacobian matrix of partials of f.

here we must asume f is a differentiable isomorphism, and orientation preserving, or else we must take the absolute value of the determinants.

in the special case that f is itself linear, this says that the area of f(U), equals the area of U multiplied by the determinant of f.