psie

ok, thank you!

yeah, should have included that. Rudin's definition is $P^2=P$ (in other words, $P(Px)=Px$).

@Thorgott right, we're running into trouble with $P^2=P$, or?

true :) ok

well, they do have the same range, I suspect

@BenSteffan ok 👍 so for instance, if $u_1=e_1$ and $u_2=e_2$ (the standard basis vectors), then $P_1(c_1e_1+c_2e_2)=c_1e_1$ and $P_2(c_1v_1+c_2e_2)=c_1v_1$ where $v_1=2e_1$, are different projections, right?

yes

It seems like his definition of vector space is a subspace of Euclidean space :)

ok

hmm, ok

@BenSteffan by ground field, do you mean e.g. $\mathbb R$?

Consider a finite-dimensional vector space $X$. Let $X_1$ be a subspace of $X$. Then there's a projection $P$ in $X$ with range $X_1$. I get the existence of a such a $P$. Let $\{u_1,\ldots,u_n\}$ be a basis of $X$ such that $\{u_1,\ldots,u_k\}$ is a basis for $X_1$ (here $k<n$). Then put $$P(c_1u_1+\cdots+c_nu_n)=c_1u_1+\cdots+c_ku_k.$$ Rudin then claims that if $0 < \mathrm{dim} X1 < \mathrm{dim} X$, there are infinitely many projections in $X$ with range $X_1$. Why?

He has introduced the Jacobian matrix. I don't know if that counts as proving that statement which I'm after.

But we need to know that $f:\mathbb R^n\to\mathbb R^m$ is differentiable iff each of its components are first, then we can apply the statement about parameter dependent matrices and continuity.

He does have a statement concerning the continuity of a parameter dependent matrix; it's continuous iff its entries are.

I struggle with understanding the assumptions of the converse with what Rudin has established prior to this theorem. Why does it suffice to consider $m=1$? Nowhere has he proven yet that $f:\mathbb R^n\to\mathbb R^m$ is differentiable iff each of its components are. I think this is key in his assumption in this theorem, or?

alright, thanks for the comments

@SoumikMukherjee which edition of Friedberg et al.? :) they have a couple of editions

Like did you read the full book or just some chapters?

@SoumikMukherjee ok 👍 thanks for sharing. How much of these books did you read?

Side note; I'm not very familiar with Axler's treatment as a whole, so my claim that he is not really using them may be inaccurate. I'm only familiar with his formatting really :) and I think it is different from other math texts to say the least.

I may have asked something similar before, but just out of curiosity, from which book did you'all learn linear algebra from? Would you recommend the book you learnt from? Second, do you think a treatment of determinants like in Axler's book is pedagogically sound (i.e. putting them in the very last chapter and not really using them as a means to prove some of the theorems in a second course in linear algebra)?

@XanderHenderson the logarithm is defined to be the inverse of the exponential function $E$, which in turn is defined in terms of its power series. So the logarithm $L$ is the function that satisfies $$E(L(y))=y,\quad y>0.$$This is Rudin's definition. He then derives through the chain rule the identity $$L(y)=\int_1^y\frac{dx}{x}$$and comments that sometimes this is taken to be the starting point of the theory of the logarithm.

Is it circular to use L'Hospital's rule when computing $$\lim_{x\to0}\frac{\log(x+1)}{x}?$$

@psie I've been stuck for hours now :( does anyone have a hunch, just a hunch, of what it means for $\psi_x(s)$ to converge uniformly to $\psi(s)$, where $x$ varies continuously, not just in the naturals?

@psie could one simply say that if $$\sup_{s\in[-A,A]}\left|e^{-s^2}-\psi_x(s)\right|\to0$$as $x\to\infty$, then the convergence is uniform?

It's causing me some confusion that we have $\psi_x$ and not $\psi_n$, where $n$ is a natural number.

I think I know which theorem in the book he uses to justify this claim, however, that theorem deals with sequences of functions, whereas here it seems we have a function that is indexed by a continuous variable $x\in (0,\infty)$. How can I deal with this?

In Rudin's PMA, when he proves the Stirling formula, i.e. $$\lim_{x\to\infty}\frac{\Gamma(x+1)}{(x/e)^x\sqrt{2\pi x}}=1,$$ he defines a continuous function $h(u)$ such that $h(u)\to\infty$ as $u\to -1$ and $h(u)\to 0$ as $u\to\infty$. Then he derives an integral expression for $\Gamma(x+1)$ in terms of the function $\psi_x(s)$ defined in the screenshot. He claims this function converges uniformly on $[-A,A]$ for every $A<\infty$.

ok 👍

yeah he probably is

Usually when you show uniqueness of something, you assume two things exist that satisfy in this case (a), (b) and (c), and then you show they equal.

This proof strategy is new to me. Showing something is a limit of some sequence to show uniqueness...hmm, maybe I'll have that for breakfast tomorrow. Let's see.

Yes, you're right :)

Ah ok, so he is showing that the function $f$ is the limit of a sequence of functions, and by the uniqueness of limits, there can only be one limiting function $f$.

@XanderHenderson that shows existence, right? Not uniqueness.

Could someone explain the conclusion of this proof? Rudin wants to show a function satisfying (a), (b) and (c) is uniquely determined. Then at the end, he just says $\varphi(x)$ is determined. What does he mean?

ok 👍

ok 👍

@User1865345 awesome, thank you. Maybe this is a tough and too broad of a question, but is the linear algebra in statistics mainly concerned with finite-dimensional vector spaces? I've heard Axler's book being criticized for being very functional analysis oriented, i.e. generalizing very quickly to infinite-dimensional spaces. Hence I am considering focusing on a text that maybe does not generalize as quickly.

I get the impression Axler's Linear Algebra Done Right is a popular choice among mathematicians. Is this also a book that anyone of you would recommend?

My situation is as follows; I have some good lecture notes in linear algebra, but they don't really serve as a reference. I mean not necessarily statistics oriented linear algebra, but I also don't mean to exclude those that are statistics oriented. I am simply looking for a good reference, a book that maybe can help make statistics oriented linear algebra much easier.

Hope it's ok I ask this question here. I've seen a couple of related questions on the site, but I thought maybe this question hasn't been asked so much in chat so I'd try here. I was wondering, which linear algebra as a preparation for statistics studies would you all recommend?

Hi there,