one issue that comes to mind is the nature of the determinant. when one considers the determinant defined by the recursive definition, it seems like a highly contrived object that is difficult to work with (as it is from that definition!). avoiding that confusion requires that a lot more scaffolding be built (ala Axler in the "Done Right" book). either way you have some work: either to untangle the meaning of the weird determinant or get to the place where you can understand the determinant as the product of the eigenvalues.
Do you know why introductory textbooks don't define the determinant in terms of the exterior product? This is how some "real" mathematicians I've talked to define it. It also is more intuitive (in my opinion) to define determinants as "signed volumes" than some sum of products multiplied by signs of cycles.
The product of eigenvalues definition is also somewhat intuitive to me ("How much does the matrix scale vectors in each direction? Now multiply those numbers together."), but it's harder to motivate the fact that adding rows together doesn't change the determinant, which is kind of important to actually computing the determinant.
it really depends on what you mean by rich: the surest path to end up with $2-5M over ~10 years is job at ~FAANG, do it well to get promoted, and manage your savings/investments well. that path is very unlikely to get you to $10-100M in the same 10 years, and starting a startup seems to be one of the best ways to do that.
According to levels.fyi, E5 at Google gets you $385k TC annually. If you save a third of that (which is not easy) for 10 years you get to approximately $1.3 million. Maybe your investments get you above $2 million, but it would really depend on what decade that was. If you had the Great Recession in the middle of that, not so great.
If you think it is easy to save more than a third of your income, remember you’ll have a federal marginal tax rate of 35% or higher, will probably have to work onsite in a HCOL state, and unless you’re lucky enough to live in WA, a high state income tax. Yes you can shield some of your income via 401ks and Roths, but for the former you’re going to get taxed on withdrawal, and for the latter you get penalized if you touch it before you’re 59.5.
Now if you’re dual cogs with no kids, maybe. If you do have kids, you’re not retiring until long after ten years.
This is very easy, actually. You just happen to like fancy things and houses, probably.
> Now if you’re dual cogs with no kids, maybe. If you do have kids, you’re not retiring until long after ten years.
I find it depressing that so many people use kids as an excuse to avoid facing their own problems with money and spending. Look deeper and you can find happiness with a lot less.
Actually I just looked at my last year at a FAANG in California.
I earned about $370 K (including 401K match). $135 K went to taxes (Federal, Social Security, Medicare, State).
If I save absolutely everything left ($235 K) for 10 years that gets me to 2.35 million.
The parent post mentioned saving $2-5 million. To get to $ 2 M I have to live off $35 K a year (I realize I'm ignoring investment activities).
EDIT: additionally to minimize taxation you’re probably doing some of that savings in a 401k. Let’s say $25k * 10 year (I was). But that actually creates a deferred tax obligation. Let’s say the tax rate you pay that at is 20% (very optimistic but by the time you pay it you’ll likely be used to living below the poverty line). That takes you down to $30k a year. Actually even worse the employer match is taxable when you withdraw it, so another deferred tax obligation means that to get to $2M (accounting for future tax obligations) in 10 years you need to live on $25K a year.
That seems challenging in Northern California. Also why would I want to do that? - living at that spending level is likely to affect my long-term health and happiness, so the motivation is not clear to me.
Of course, a bad market makes it longer, but the converse is also true: it's more likely a good market makes it shorter! (historically markets go up more than down, of course)
Add a spouse that also saves and invests, and you can have MORE than $45,000/yr of expenses covered within 7 years of earning at that rate. You can also back off to part time or have one spouse continue working after having kids, and all you need to do is cover SOME (not even all!) of your expenses.
> I want to do that? - living at that spending level is likely to affect my long-term health and happiness
The freedom of not dealing with "the cult of impact" [1] and other such silly things is amazing. Having a huge pile of money allows solid peace of mind in a way most people can't even conceive.
And finding out how much happiness you can get independently of spending money is truly eye-opening. Most people are too scared to even TRY finding fulfillment away from what society tells us is "fun".
> But that actually creates a deferred tax obligation. Let’s say the tax rate you pay that at is 20% (very optimistic
There are also ways to get at this money with almost no taxes paid later IF you are not working--look for "Roth IRA conversion ladder".
Also look into "Mega Backdoor Roth" for more tax sheltering.
> That seems challenging in Northern California
You don't have to retire to California. Just pay minimal costs while you're there. Many places in US are beautiful, have great infrastructure and health care, and don't cost as much.
And investment growth is a LOT; don't ignore it. And after just a few years of not working, unless you have a bad start you can probably live off of much more than the initial safe withdrawal rate.
i was going to sit this one out, but i want to +1 the above with more than just an upvote. Austen seems to be arguing and explaining all the happenings in good faith with someone who appears to be hell-bent on "catching" him with something by misconstruing things.
i have never understood why Lambda/Bloomtech has so many haters but the other commenter here appears to me to be in that camp.
Austen is a terrible person and has been caught lying dozens of times and always hides behind the "we're just good people trying to help others" line to distract from the fact he and the rest of the lambda school staff (which is just all of the un-hired grads so they can boost their already pathetic placement rates) are scammers and frauds.
Calling someone a terrible person, liar, scammer and fraud with little evidence is not a good look. Even if all your points are true (Austen has refuted at least a few), "terrible person" is a huge stretch.
this is all accurate and i'd add that it is deceivingly hard to make something that sustains itself via advertising. when youtube was acquired google had a large part of that already figured out.
one of the things that helps generate new ideas that can be cultivated is the ability to be playful. once a given problem or subject is sufficiently loaded onto a brain, if that person can relax and have child-like naivete about poking and prodding, novel insight is usually not far.
cultivating this ability is fairly well understood in a lot of domains, i think. two examples that are top-of-mind are improv and jazz.
> The author of this article was paid $42 million last year. Uber's top 7 executives received: "$11.4 million in salary and cash bonus, plus $71 million worth of equity awards."
it's misleading to call equity compensation "getting paid". there's a strong argument that executives are being over-payed, but say the CEO receiving a salary of ~1.6M (11.4/7) is a lot less egregious than them being paid $42M. surely the equity conferred to the execs gets them some cash availability in various ways, but it isn't exactly the same as just giving them money (eg what would happen if the CEO uber sold off all of his stock?).
maybe the cash portion of exec salaries ought to be much lower or maybe their equity compensation ought to be much lower, but i don't think either of these things by itself or in combination is going to solve the structural problems of the american economy such as healthcare being tied to employment.
> it's misleading to call equity compensation "getting paid".
Please.
A person who has a portfolio worth tens/hundreds of millions of dollars does not have to liquidate it to leverage its value. They can secure cash loans on extremely preferential terms backed by their assets, for one.
Futher, who cares if it's "exactly the same as giving them money" or not? That doesn't change the inequality inherit in such compensation packages, just how they pay taxes.
> [neither reducing cash or equity comp] is going to
> solve the structural problems of the american economy
> such as healthcare being tied to employment.
Sure, that's true, but that's also a fine way to allow wealth inequality to continue unchecked. Jeff Bezos can't be worth over $100B unless his company is vastly underpaying and mistreating its workers.
Amazon has been paying Jeff Bezos the same $81,840 salary since 1998. On file it says he gets about $1.6 million per year, but that's mostly security costs. He doesn't get any bonuses and he doesn't get any more equity. So none of the profits that Amazon has been making have been going towards Jeff Bezos pocket to make him richer. He's worth what he is because he owned something like 43% of Amazon at IPO.
If Jeff Bezos quit Amazon in 1998 after the IPO and we were here today at the same > $3000 per share price of Amazon, Jeff Bezos would still be worth over $100 billion and would have had nothing to do with Amazon for the last 22 years. So I don't see how you can draw the conclusion that Jeff Bezos can't be worth $100 billion unless they're mistreating their workers. Those things don't have to be connected.
Now should Jeff Bezos be worth that much? Probably not. But he's worth that much because he owned a large stake in Amazon. Anyone could have purchased Amazon at IPO and would have been up over 2000x right now. If you put $500k into Amazon at IPO you'd also be a billionaire right now, does that mean that Amazon is mistreating their employees because of you?
> So I don't see how you can draw the conclusion that
> Jeff Bezos can't be worth $100 billion unless they're
> mistreating their workers. Those things don't have to
> be connected.
But...they are connected.
Bezos is not just a large shareholder of AMZN, he's also it's CEO. He is responsible for every decision, ultimately, such as warehouse worker count and compensation. If they make less, he can put what he would have paid them into other product lines, thereby increasing AMZN stock price and his personal wealth.
Jeff Bezos could choose to pay his employees a living wage and treat them with dignity, but that may impact AMZN's bottom line or its share prices, so he doesn't.
> If you put $500k into Amazon at IPO you'd also be a
> billionaire right now, does that mean that Amazon is
> mistreating their employees because of you?
This is a stupid hypothetical. I wouldn't be the CEO, so the decision wouldn't be mine. And I'm assuming I somehow didn't buy a significant enough sum of stock to be able to influence shareholder votes.
Stop defending billionaires. They won't defend you.
The comments in this subthread are talking about CEO salary. The point being, Bezos isn't rich because of his salary, he's rich because he owns shares, so adjusting his compensation isn't going to make a meaningful impact to the prosperity of Amazon employees. (Managing to effectively tax the corporation might though...)
I always find myself in the weird position, that i think, i have to defend this kind of wealth. I do not think, that wealth inequality gets fixed by criticizing it, while at the same time i am using the services, that Amazon offers and has benefited my life. The same for by the whole world. By millions of people and companies. I paid for this wealth inequality. I take the full blame. I would switch without hesitation to something else, if it is better.
The solution for me is not to redistribute wealth by force from someone, whom i have paid before for a great service. That does not compute for me. It feels like a not justifiable punishment.
So for me the question is rather, what are the alternatives. Am i happy with AWS? Of course not. Do i want people to deliver packages and get paid badly? No. Do i find automatic package delivery attractive? Yes. But then driver lose their jobs. How to deal with that.
How to create wealth on the new foundations, where more people can participate? And dont i already see signs of it?
And, skilled labour is always wanted. All the non-computer, but crafty people do well. Everyone who can build something, that others want and cant build it, bcs... they do other things for others.
Ah, but its a useless rant.
Maybe the US is doing the right thing, bringing back the production side of things back home, so this kind of margin will be closed. Maybe it will increase then even more the productivity to make up for it. Who knows.
> I always find myself in the weird position, that i
> think, i have to defend this kind of wealth.
You can easily just...not.
To continue using Bezos as an example: sure, he built a logistical empire that can deliver cat litter to my home in under 48hrs, and a computing empire that's surprisingly well used despite being absolutely obtuse.
If that's all he did, and everyone who worked for Amazon was treated well, made middle-class salaries without overworking themselves, and had world-class healthcare? Sure. I got no complaints.
But that isn't what's happening. Wealth inequality isn't "I'm jealous of someone" or "I'm opposed to this just because" -- it's recognizing that it only seems to happen when the folks on the bottom of the org chart suffer.
There does appear to be a push away from globalization and towards isolationism. However, this may not last, and even if so, it remains to be seen whether the government will be able to effectively encourage corporations to move manufacturing back on-shore.
Jeff Bezos is a bad example for the point you're trying to make. He's worth that because he held (and built) an asset that increased massively in value over the past ~30 years. Not because his comp was too high.
Couldn’t his company be worth more than a trillion because it’s...adding far greater value to the ecosystem it exists in? And his share is in the ~100B as he captures the upside through being a founder and taking the early risk?
But he didn't? Yer man Travis was the founder, this dude Dara is just a career executive brought in to turn around the ship. He basically hasn't taken any risks, with respect to Uber at least.
> it's misleading to call equity compensation "getting paid".
how strange, I think it's misleading to not call equity compensation "getting paid" and the IRS seems to agree? When my company grants me vested shares, a whole bunch go right to the IRS instead. Because it's "income".
This is kind of a red herring because while you’re right that it would be a bad sign for a CEO to dump all their stock, they have access to whatever liquidity they need through it, and are likely going to diversify their gains.
I agree, equity is likely even more valuable then the price given. It gives part ownership and voting rights in Uber. The employees could collectively use their voting rights to have a voice in the direction of the company.
The equity gives them a very strong position to make many many millions more. Yeah it's more complicated to cash out, but it's a huge opportunity not granted to the labor.
i agree wholeheartedly with the spirit of your comment, but saying "nutritional science is complex" or "the dosage alone makes it so that a thing is not a poison" does not help people figure out what foods to choose or how much to consume.
looking at the glycemic index is a useful heuristic and some of the research on canola oil makes me skeptical of it, and i think it is good for the author to point that out, as it suggests that oat milk might not be as safe as it is marketed to be.
another heuristic is biasing in favor of food that is processed less eg eating a bowl of oats is probably a better idea than drinking the analogous amount of oatly. similarly, it is a good idea to eat foods that people like you have been eating for long times, which in the oats vs oatly example favors the oats.
the precautionary principle suggests that the onus is to verify the safety of a given new food, not to prove that it is unsafe.
The one thing I'd change there is to delete the word "new" from the last sentence.
There are all sorts of foods that are terrible from a health perspective, but get grandfathered into a culture because they became popular before science really caught on. If you're closely analyzing the biochemistry of oatly or soy milk or whatever, and finger-wagging at others for drinking them, but still drink beer on a regular basis, you may be engaging in premature optimization.
Or, just looking at sugars: A serving of Oatly does contain 7g of sugar. The same amount of whole milk contains 13. You don't notice it because lactose tastes less sweet than many other sugars, but it's still a pretty generous dose of calories in the form of simple carbohydrates. I can't personally make the glycemic index comparison, because, while I can guess, I have no actual idea what the glycemic index of two tablespoons of lactose would be. Or even if measuring that would tell you anything about the glycemic index of a complex foodstuff that happens to contain that much lactose. I suspect you can't actually directly infer the one from the other, so, unlike the author of TFA, I think I might choose to proceed with caution there.
As far as what to consume: The messaging out there is remarkably consistent, as long as you tune out any information that's being provided by people who are trying to sell you something. Be it an actual foodstuff, a book, fitness lessons, or even just advertising. Even Michael Pollan nailed it pretty early in his career. In a single sentence, too. It's just that then he had to keep going, because stopping there wouldn't have made for much of a career as an author.
> There are all sorts of foods that are terrible from a health perspective, but get grandfathered into a culture because they became popular before science really caught on.
Yes and no. There is some element of an inherent scientific method involved in having cultures eat various foods for tens of thousands of years and then seeing which ones are still around at the end of the “experiment”
Perhaps there's some effect there, but this idea that more competitively successful cultures have healthier diets is difficult to reconcile with a lot of other memes around health food.
One also has to reckon with the fact that people's health needs were very different a couple hundred years ago. Once upon a time, the greatest health concern was getting enough nutrition. Also, when your life expectancy is 30, you probably don't give a damn that, if by some chance you live to see 70, all those preserved foods you used to get through the winter will end up having increased the chance that the thing that eventually kills you is stomach cancer.
As TFA's hand-wringing about sugar clearly indicates, though, we're just not playing the same game our ancestors did. The major worry nowadays that it is now, uniquely in history, possible for almost anyone's primary health concern to be that they're getting too much of a good thing.
Surely the species of most foods available nowadays are only a couple centuries or decades old?
Plus there's survivor bias: just because population A survived on food X, doesn't mean it won't halve the life expectancy when introduced to population B. E.g. high calory foods in the oceania populations caused obesity.
i was a physics PhD student once and i might have some thoughts on how to learn physics, but i think the parent commenter is spot on -- the only reliable way we have is to have an apprenticeship with someone who works well with you.
there are a tons of posts along the lines of this one, which are largely lists of books that people like to romanticize or as the parent comment says, lists of things that someone thinks you should know.
the Feynman lectures are a good example of that, actually. i've read them a couple of times and they are wonderful -- like most things Feynman. but very unhelpful. even Feynman was disappointed at how the students they were given to weren't quite learning the physics. Feynman is to explaining science as, eg Joshua Bell is to performing music for the violin. but in terms of actually trying to learn what a physical theory says, to what degree it holds, what are the current open problems with it, and what might be good approaches to solving them, Feynman explanations outside of highly technical works (like his papers or maybe his lecture notes on statistical mechanics) are not too useful. and the more technical works are very difficult to approach by yourself, you'll need a mentor.
