Thank you all for your input. So it happens that I typed my argument into Microsoft Word and when I copy and pasted it, the revisions I made were incorrect. For example, the set S should have read x^n not x_n. My wording was a bit poor at times, by scalar addition I was implying a faster method...
Thanks for the explanation! I wrote it in my own words to make sure I'm grasping the root of problem properly. Mind confirming me?All functions produce an image space. This space has a dimension, based on the series of functions f_n(x).
Where S = { f_n(x)=x_n | n ∈ ℕ } for all x in [0, 1]. S is...
I was hoping you guys could help me in understanding some vector spaces of infinite dimension. My professor briefloy touched n them (class on linear algebra), but moved on rather quickly since they are not our primary focus.
He gave me the example of the closed unit interval where f(x) is...
Great! Thanks so much.
To briefly expand on your explanation, when you say it sends the basis (by the linear transform), how exactly do these numbers come about mathematically. I know I may be overlooking something, but I want to be complete.
From what I see, the length of the basis is clearly...
Ya thanks! Now that you have me thinking I'd like to understand exactly how I arrived at this solution. I got that equation (or matrix) from my text: Linear Algebra 4e by Bretscher. Book is VERY heavy on application, thus my struggle with the general theory.
Let A=[a1 a2 ; a3 a4]. since y=x the vector x=(x1, x2)^t=(1, 1)^t
From the definition of a projection= [u1^2, u1*u2 ; u1*u2 u2^2]
I can see that if a1=a2=a3=a4=1/2 then,AA=A. However, I'm not quite sure on how/why?
Edit:
From Ax=b let x=(1 1)^t and A=projection and solve for u1 and u2 (which...
... I was wondering as in I cannot recall or figure it out.
Reducing it to algebra in terms of some a1 a2 a3 a4 (forming a 2x2), all that resolves is trivial solutions.
Would it be that difficult to show me one?
To simplify this inquiry let us deal with R2. I know that if you have the identity matrix (let that be A), then AA=A. However, I recall the existence of a matrix with all nonzero entries that had the same property. Thinking of rotations, I cannot think of one, (since rotating by 0, or 2pi is...
Ya I'm not quite seeing how it works... it is probably due to my own error, let me elaborate.
taking the natural log and pulling down the n we get n * ln( 1 + (1/ n^2) ).
manipulating the n we get ln( 1 + (1/n^2) ) / (1/n)
applying L'Hopital's rule we get lim [(1/n^2) / ( 1 + (1/n^2) )] /...
I see. I just wanted some clarity on how the sequence acts when the degree of n in the (1/n) term is greater than the degree of the entire sequence. Is it fair to say that ( 1 + (a/n^2) ) ^ n converges to 1 as well, regardless of a?
(Thanks for looking into it!)
I was thinking of how ( 1 + (1/n) ) ^ n converges to e and I am aware of how if it is raised to some an, then it converges to e^a. If i recall if the form ( 1 + (a/n) ) ^ n converges to ae?
I was hoping someone could tell me how to deal with ( 1 + (1/n^2) ) ^ n?
Thanks!
Well if it uniformly converges from 1/2 to infinity then using the min of 1/2 (bc all others from 1/2 to 1 will follow suit), g_(1/2)=(1/2)lim(n/exp(nx)) for large enough n, this term goes to 0. So does this conclude that it uniformly converges on (0,infinity)? Not including 0 since the...
My reasoning for my guess was that it would converge up until it reaches its maximum, which is generally the case for problems in my course (this is the first time I've encountered a function of this complexity, hence my confusion).
Since the maximum is at x=1/n, each x value beyond x=1/n would...
I see! Thank you for being so patient with me lol. So as for the second part of the original question, is there any interval in which it converges uniformly? Is it where x is [0,1].