Recent content by VrhoZna

I see now. If ##y \in A## and ##c \leq y##, then X is a subset of the chain ##X \cup \{y\}## and so ##X = X \cup \{y\}## and ##y \in X## and we must have y = c.

Homework Statement From Introduction to Set Theory Chapter 8.1 exercise 1.4 Prove that Zorn's Lemma is equivalent to the following statement: For all ##(A,\leq)##, the set of all chains of ##(A,\leq)## has an ##\subseteq##-maximal element.[/B]Homework Equations N/A The Attempt at a Solution...

Try expressing the powers and the coefficients in a fractional form while taking derivatives and see if the pattern becomes clearer that way.

I typically listen to white noise exclusively when studying anything like math or physics but I like downtempo instrumental hip hop whenever I don't want to listen to TV static.

If you mean the definition of the annihilator the book gives it as, (I wasn't sure what exactly you meant by explicitly defining it) "If V is a vector space over the field F and S is a subset of V, the annihilator of S is the set ##S^0## of linear functionals f on V such that ##f(\alpha) = 0##...

You shouldn't use approximations as they don't have a place in mathematical problems of this nature and they don't mean anything without specifying a tolerance for error. sin(pi) is approximately equal to pi like 5 is approximately equal to 2 The limit I was referring to is lim x --> 0 (sinx/x)...

No, reckless assumption often leads to errors. Do you recall that special trigonometric limit that involves sin? It involves a limit where x tends to 0 but with a clever use of substitution it may help you.

Why not just try out that same method? What can you compare sin2(1/x) to?

Woops, for the last part to show ##ann(V_i) \subseteq Im(^tE_i)## let ##g \in ann(V_i)##, ##\alpha \in V## and suppose ##\alpha = \alpha_1 + \cdots + \alpha_k## with ##\alpha_i \in W_i## for i = 1, . . . , k. As ##^tE_i## is a projection we must show that ##g(E_i(\alpha)) = g(\alpha)##. We have...

Perhaps, but other than a quick wikipedia article read just now I haven't learned anything about quotient spaces so I'm not entirely sure what you're pointing out. As for the second part, would defining the annihilator of one of the subspaces ##V_i## as ##\{ f \in V^* | f(E_i(\alpha)) = 0 for...

I mean more in a self-study sense than required reading for a course. Currently I'm working through 4 books, Calculus Vol 2 by Apostol Linear Algebra by Hoffman and Kunze Introduction to Set Theory by Karel Hrbacek and Thomas Jech Learning the Linux Command Line by William Shotts (I'm an Arch...

Mental arithmetic... I always lose a sign somewhere.

Not in general, consider the dot products of two non-parallel vectors with the 0 vector.

(From Hoffman and Kunze, Linear Algebra: Chapter 6.7, Exercise 11.) Note that ##V_j^0## means the annihilator of the space ##V_j##. V* means the dual space of V. 1. Homework Statement Let V be a vector space, Let ##W_1 , \cdots , W_k## be subspaces of V, and let $$V_j = W_1 + \cdots + W_{j-1}...

I suppose I hadn't realized the full implications of said subfields having characteristic zero at the time of writing the proof, but I understand a bit better now. Thank you for your answer.