matt grime

yes, that is the idea of it, if you know where it sends the basis you know ehrere it sends every vector.

extending the sigma field to a larger one simply means we have more possible outcomes to which we assign probabilities. in you case here it would seem the expectation depends on the sigma field since the sigma field tells us what the probabilites of any events are.

Trivial? Most poeple find them very difficult to begin with. The reason is to give a solid mathematical proof to the idea of continuity, better than the (incorrect) notion that sometinhg is continuous if you can draw its graph without taking your pen off the paper. Firstly, how do you know how to draw a graph ofs omething complicated? secondly this only applies to fucntions of one variable y=f(x), and do not tell us about the fucntions that we meet in the 'real world'. All it is is saying that we know y=f(x) (and these could be vectors now) and suppose we're allowed an error in measuring y (we always have errors) then how close to x must i make s so that f(s) is allowably close to y? The epsilon delta argument is just that set out properly. OK, outside of a mthas course you're nto going to prove anything is or isnt' continuous, but you need to know what continuity means and what it implies. sadly a lot of teachers do not give the correct interpretation of continuity, why is a mystery. i generally explain it as a game. given f some function then we'll play a game. take x and f(x). you pick some e and if i can now pick a d so that for all y d within d of x f(y) is within e of f(x) then i win, otherwise you win. the points where i win f is continuous and if i win at all points x then f itself is continuous. another way of saying the same thing is the f keeps things that a re close enough together close enough together. example: define f(x) on the rationals, f=0 if x is lesss than sqrt(2) 1 if x is greater than sqrt(2) this function is continuous even though it appears to have a jump at x=sqrt(2) but of course that is an irrational number. in any case the best way to deal with continuity to begin with is that f is continuous at x if all sequences x_n tending to x satisfy f(x_n) tends to f(x)

i think you need to reword your question as it makes little sense. The word 'information' appears out of place. In probability (and measure) given some set W a sigma filed S on W is a set of possible outcomes from sampling from W to which we can assign probabilities.

Construct one that does it and the uniqueness follows (a liner map is uniquely determined by where it sends a basis and specifying the action on the basis uniquely determines the map obviously). do you understand the basic method of constructing linear maps from where they send the basis vecotrs? eg in R^2 if e and f are the basis and I tell you that e maps to 2e+f and f maps to e+2f can you write out the matrix? (you can if you just think about matrix multiplication. where does the 2x2 matrix ith entries a,b in the top row and c,d in the bottom row send the vector (0,1)^t (ie the column vector with 1 at the top and 0 at the bottom) in terms of a,b,c, adn d?

you don't need to bound anything with + in it, my mistake; i read theough the post too quickly. so how did you bound x.x_n?

a function is continuous if it is continuous at every point. therefore to show it is not continuous you need to find one point where it is not continuous. please, forget the raising your pen nonsense.

i, j, they're just dummy indieces, i could have used, s,t,r, or "squirrel", you get that it doesn't matter?

Don't worry, I was trying to amke a point: part of the definition of function is its domain; seems like you have a poorly written maths book (don't worry, most of them are). In this sense just remember that you can only take square roots of positive numbers. If you don't know what it's asking though, how do you expect ot find the answer? what is the definition of continuous?

defined on what?

Now you need to get a bound for |x+x_n|, what can yuo do with that? why do you have a cross in there?

Please don't take this as rude, but what is yor thinking behind that step? Mightn't it be more helpful to think about how the w_i relate to the v_j?

I quite liked something (a fictional conversation) that Miles Kington wrote in the INdependent this week. It contained the clincher argument against ID: any designer who designed the life we see around us can in no way be called intelligent. Or as Joseph Heller wrote, approximately, how much respect can you have for a asupreme being the chose to include tooth decay in his grand design? I forget the exact words but that's it in spirit. Complexity in a design is a sign of stupidity not intelligence.

if you don't know what the minimal and characteristic poly are (in relation to one another) you shold go and look them up in you course notes. and as i say somewhere else in reply to this, the second part is true becuase of the definitions of an inner product on a complex space, something you need to look up, again in your notes. maths is infinitely easier if you know the definitions.

what is the definition of minimal polynomial? what can you say about it with respect to the characteristic poly? and if you know that an inner product is a (complex conjugate) bilinear form then what's the problem with the second part? I mean waht is ||x||^2 in terms of the inner product?

And here we go again. Firstly let me make clear that I am about to explain the way infinity appears in mathematical literature. If you want a layman's interpretation you are in the wrong place. There is no one such thing as a simple unqualified infinity in mathematics. It is better instead to think of things as being finite or infinite where infinite means not finite. In general we use finite as an adjective to mean that some measurement of some object is an element of the real numbers. In terms of sets, where we have discrete elements then a set is finite if it contains a finite number of elements in the usual sense (ie its cardinality is an integer). If a set is not finite then it is infinite. This is the first time we meet the use of the word infinity when people say such things as there is an infinity of integers. Now, in maths Cantor came up with the infinite cardinals, these are attempts to compare sets with an infinite number of elements. For instance we know that a set with 4 elements has fewer elements than a set with 5 elements. But how do we make this rigorous with infinite sets? He chose to use the distinction that two sets *wil be declared* to have the same cardinality if there is a bijection between the two. More easily understood is perhaps that S and T have the same cardinality if there is an invertible map/function from S to T. In this sense we have aleph-0 as the first infinite cardinal (it is used to describe sets with the same cardinality as the natural numbers). Sets that have this cardinality are the integers (ie the natural numbers and their negatives) and the rational numbers. In this sense we can define an addition on cardinals, and in this sense 'infinity plus one equals infinity' or if X is any infinite cardinal then X=X+1. For example the natural numbers and the natural numbers with zero have the same cardinality yet one has 'one more element than the other'. The bijection is given by [math]f:\mathbb{N}to \mathbb{N}\cup\{0\}[/math] where f(n)=n-1, its inverse is g(m)=m+1. It should be noted that there are infinte ordinals too wherein adding 1 to an infinite ordinal produces a different ordinal. Another place in infinity is the more usual one of 'something larger than any real number' so it is simply a symbol that we can add to the real numbers and we can think of it as being a useful symbol for doing analysis: something tends to infinity if it grows without bound. We may also add in negative infinity as smaller than all real numbers. we can now extend certain algebraic operations to infinity (but not all) from the real numbers. and again in this case infinity plus one is infinity, since if we are to have infinity plus one a sensible number in this extended system it must be larger than all real numbers, and the unique element larger than all real numbers is infinity. note again that there are extensions of the reals (called hyperreals or surreals depending on whether you prefer robinson's or conway's flavour) where there are 'infinite' numbers and adding one to an infinite number produces a disticnt infinite number.

do you understand the idea of 'negation of a proposition'? because if this were true then its converse would be false, and the converse is that all sets of fewer than p vectors in a space of dimension p are linearly independent, which is obviosuly completely, utterly, and trivially false. (eg v, 2v for any nonzero vector v in a space of dimension 3 or more. and even more trivally, any set with the zero vector in is linearly dependent.))

because the series you can't sum r^2/3^n you replace by one you can, (2/3)^n and you compare the (easily evaluated) sum of the tail of one and which by the comparison theorem we know is more than the sum of the tail of the other.

but it is an important point: do you understand the difference between "if", "only if", and "if and only if"? sometimes they use sufficient, necessary, and necessary and sufficient.

sorry, but you aren't correcting me. i didnt' say that r>5 was neccesary, only sufficient.

certainly it is not correct.

mmm? you want an N such that [math] \sum_{r=N+1}^{\infty} r^2/3^r[/math] is less than 10^{-6} Well, that sum is less than [math]\sum_{r=N+1}^{\infty}(2/3)^r[/math] for N sufficiently large (ie when 2^r>r^2 which is true for all r>5 certainly) and that sum equals 3(2/3)^{N+1} so pick N such that that is less than than 10^{-6} say any N greater than 100 would do

when you understand it you will see that you are correct; you need to stop being so concerned with unimportant things like this particular question and start to think of the bigger reasons so that you understand why you are correct. if you don't learn why you are correct you will never learn when you are correct and you'll continue to ask us to verify your work. at some point we should not need to do that for a subject. What is a basis? It is a linearly independent spanning set. 1. every basis has the same number of vectors in it, this number is the dimension of the space, let d be the generic letter for dimension. 2. all linearly indepedent set have at most d vectors in it. 3. any linearly independent set can be completed to a basis. 4. d is the maximum number of vectors that can be in a linearly indepedent set. 5. put it all together.

it may not say it explicitly since it is immediate from the definition of dimension: the maximum cardinality of a linearly independent set, or the minimal cardinality of a spanning set. any linearly independent set can be completed to a basis, so if the cardinality is already maximal (ie no more elements can be added) then it must be a basis.

analysis, that's why it's in the analysis/calculus section. small irony in kidia being answered by radiohead, just one letter.... anyway, the question, if i insert brackets, becomes find the limit of [math]\frac{x+y}{x^2+y^2+1}[/math] as (x,y) tends to (0,0). call that fraction f(x,y), clearly |f(x,y)| < |x+y| since f(x,y) is just x+y divided by something bigger than 1 and if i take any positive Z, and divide by k where k>1 tehn Z/k< Z. Now, suppose that |(x,y)| < d, ie (x,y) lies inside the circle of radius d centred on the origin in R^2. what is the largest value of |x| and the largest value of |y|? note i don't need to find the largest value of |x+y| since I can always overkill things in analysis; estimates are good enough if they work (slight tautology, sorry). but here |f(x,y)|<|x+y| < |x|+|y| (triangle inequality) < something you should have found.. now, given e>0 pick d such that the "something you should have found" is less than e. this is the standard way to do all e-d proofs:suppose the input to be less than d, use this to hopefully show that the ouptut is less than something dependent on d that can be made arbitrarily small, ie smaller then e.