Recent content by aeroplane

You are right, there is an infinite number of solutions, solving for x gives you your answer or pi/6, and so the tangent of y=2(x-cosx) is parallel to 3x-y=5 whenever x=pi/6 +2pi*n where n is an integer. As said above, instead of considering the intersection of the two functions, you are...

Is it necessary to solve this using integration by parts? There's a nice substitution that makes the integral straightforward. I couldn't easily see a nice way to separate the integral.

From what I can tell {v_{1},v_{2},...,v_{k}} is simply a finite subset of V, and {T(v_{1}),T(v_{2}),...,T(v_{k})} is the corresponding subset of images in W. As such, I don't think this question concerns bases of either V or W. That said, when a set {v_{1},v_{2},...,v_{k}} is linearly...

I don't think the matrix you have is a vector, vectors are either row vectors (1xm matrices) or column vectors (nx1 matrices). For this system of linear equations, each row represents a different equation. The system is then linearly independent if each row is independent, ie. row reduction...

From http://en.wikipedia.org/wiki/Limit_point we have that x is a limit point of A={sin(n): n a positive integer} if every neighborhood of x has another element of A different from x. So if you can show that you can approximate x as closely as you wish with elements taken from A, then you have...

Definitely Deveno, but it's good to understand that it is an infinite product, otherwise you lose the uniqueness of each factorization.

Sure: we can write 108 as 108=(2^2)(3^3)(5^0)(7^0)... and 405 as 405=(2^0)(3^4)(5^1)(7^0)(11^0)... . Using our formula for (108,405)=(2^min{2,0})(3^min{3,4})(5^min{0,1})(7^min{0,0})(11^min{0,0})... we see that (108,405)=(2^0)(3^3)(5^0)(7^0)(11^0)... = 9. This divides both: 108/9 =...

Sorry, some other arbitrary set, S={s_1,s_2,...,s_n}, S'={s'_1,...,s'_n} since both are finite.

For two sets SU{x} and S'U{x} in X, SU{x}-S'U{x} is the empty set iff S=S'. Every set of X has to be at least {x}, so the difference between two elements has to be related to a difference in the subsets of S. Then you can think of X as P(S) where each element has {x} attached to it, so it must...

For 2) any element T of X is subset of K with x inside it. x isn't in S so T must be at least {x}, but it can also include elements of S since K is the union of S and {x}. If you take two elements SU{x} and S'U{x} in X, thinking about the set SU{x}-S'U{x} should help you prove the last part.

I take it (a,b) = gcd(a,b)? I'm not sure if it is just a difference in notation or not, but I think your prime factor representations should be infinite, without a index k where it stops. It's a product of all primes to different powers greater than or equal to 0, and there are an infinite...

Solving this limit may help (you can even use l'Hopital's rule!) lim x->infinity sinx/x

Have you tried making a new line y_p perpendicular to both? The segment between the two y lines gives you your perpendicular distance.

Looks good, I assume you worked out the transitivity property yourself, but if not you should show your work. xRy ^ yRz => |x|<= |y| and |y|<= |z|, and since <= is transitive (do you need to or have you shown this before?) |x|<=|z|, so xRz.

An open set like (0,n) is not compact, consider the family {C_n}={(1/n,+infinity) for n in N\{0}}. (0,n) is contained in the union, so {C_n} is a covering of (0,n). Assume it has a finite subcover. Then a finite union of C_i will cover (0,n). It's finite, so there is a largest value for i...