@TedShifrin: I only know how linear algebra was taught in my case, and I don't really know whether you would think it was not enough integrated with analysis. so maybe an example where you personally integrate it or would integrate it more?
I created a course and textbook, @Huy, that does linear algebra and multivariable calculus/analysis interwoven. Most students in the US learn multivariable calculus with no linear algebra involved at all. There are a few books that have done this integrated treatment: Williamson/Crowell/Trotter, Hubbard/Hubbard ...
@Huy: I'm rather surprised that your library has two copies. However, I know that the text has been used for courses in Britain and elsewhere in Europe.
I start with two perpendicular vectors in $\Bbb R^2$, and use similar triangles to deduce that $x_1x_2+y_1y_2=0$. Thence motivating the dot product in general.
LOL, sorry about sticking you with game theory, but I think you'll love it ...
@TedShifrin: I will start showing my high schoolers how a system of linear eqs can be written with matrices and vectors and then the scalar product (which they don't know yet) is just the product of two vectors. then I was going to prove that this has the property of being related to the angle between the two vectors. do you think that is pedagogically bad?
What I say is nowhere near offensive @skullpatrol, compared to the rest of the users here. If I say "<some branch of math> is BS" why should it deserve an ignore?
@TedShifrin: I think I will like Game theory too, I don't blame you for it at all. I really needed enough participants for the class to form, because it increases my salary for next term by 50% and otherwise I would have to be on budget
@TedShifrin: there is a lecture at my uni in spring about it too. I think I will study the book of the lecturer about Game theory, use it to prepare my course and then take the exam at uni for free
BTW, @Huy, it wasn't in the book, but I ended my course this fall with a discussion of Fourier series (an extension of orthogonal projections, just working in $C^0[-\pi,\pi]$ with the $L^2$ inner product) and applications. One was Chris'ssis's obsession. The other was the Isoperimetric Inequality in differential geometry :P
LOL, no, @Studentmath, the test was interesting enough. I'm done :P
I really hope I can get to eigenvalues fast enough with my students. I'd hate to teach them so much about linear algebra and have to skip cool applications just due to time issues ._.
I have one thing I actually still didn't really decide. Do you think talking about the rank of a matrix is necessary for high schoolers? Another teacher thought I should skip it to get to eigenvalues faster, but I feel like it can also show quickly whether a system of lineqs has no, one or infinitely many solutions (and what dimension the solution space has)
Rank is easy, @Huy, in terms of echelon form. Worrying about rigorous linear independence and bases and the four fundamental subspaces slows you down more. But I definitely want dot product and orthogonality as important ... which many linear algebra books postpone 'til the very bitter end.
Can anyone advise me as to how to write this: $[ p^*_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of partitions of }n}{\text{into an odd number of parts}}. ]$ as a power series representation?
@DanielFischer I initialized the array and I got this:
Give dimension: 9 Give y: 11 Give a value for the 0 position of the array -55 Give a value for the 1 position of the array -13 Give a value for the 2 position of the array -5 Give a value for the 3 position of the array -1 Give a value for the 4 position of the array 0 Give a value for the 5 position of the array 43 Give a value for the 6 position of the array 52 Give a value for the 7 position of the array 85 Give a value for the 8 position of the array
Which is greater, $9*12$ or $10*11$? What about $100*103$ vs $101*102$?
well consider $a, b, c, d, a<b<c<d$ and all consecutive integers. then $a(a+3) = a^2+3a$ whereas $(a+1)(a+2) = a^2 + 3a + 2$, so clearly b*c is greater. **BUT ONLY BY A VALUE OF 2**.
In fact this could be generalized to nonconsecutive integers, just take the first integer to be $a$, and then use $b, c, d$ in terms of $a$. e.g. $302*405 ?? 350*375$ means that you get $(a)(a+103) ?? (a+48)(a+73)$ to get $a^2 + 103a ?? a^2 + 121a + 3504$, clearly the second product is greater by $18a + 3504$. Interesting.
@evinda We should. You have a bug in your code. One on line 39, another is that you don't watch out for ... The other one is what gets you the wrong result here. There may be more bugs, haven't looked at everything. Time for a debugging session. The good old-fashioned printf-debugging is fairly easy and tells you what's wrong. Additionally, put spaces around your operators, that makes the code much more readable. And generally, you shouldn't expect sorted input, so you should sort your array
