Frenemies of TalkRational:
Nontheist Nexus |  Rants'n'Raves |  Secular Cafe |  Council of Ex-Muslims |  The Skeptical Zone |  rationalia |  Rational Skepticism |  Atheists Today | 
TalkRational Archive  

FAQ Rules Staff List Calendar RSS
Go Back   TalkRational Archive > Discussion > Mathematics

Notices

Mathematics constants, variables and stuff

 
 
Thread Tools Display Modes
Old 05-10-2016, 07:16 AM   #2647837  /  #1
uncool
Senior Member
Plasmatron Champion
 
Join Date: Jul 2008
Posts: 15,106
uncool
Default uncool actual school on linear algebra

So the last thread, I accidentally turned what was supposed to be a "school" on linear algebra into an exploration of linear algebra through the lens of category theory. Which was useful to me, but probably less useful to the people who were watching the thread. This thread is for those people.

So! Linear algebra.

What is linear algebra? Linear algebra is the study of what you can do when you are restricted to two basic operations - adding things together, and multiplying everything by a number.

Seems restrictive, right? Well, that's a lot of how math works - by restricting what you think you can do, you can make more situations similar - and so you can study things much more deeply.

The thing is, we actually haven't restricted ourselves all too much. Or rather, we've restricted ourselves to a pair of very useful things we can do. Don't believe me?

Every day, Alice buys two apples and one banana for breakfast. Every day, Bob buys one apple and two bananas for breakfast. Last month, Alice paid $5, while Bob paid $4. This month, Alice paid $5, while Bob paid $7.

How did the prices change?

Let the price of an apple be a dollars, and the price of a banana be b dollars. We then get the equations for last month:

2a + b = 5
a + 2b = 4

So let's solve. Double the second equation - multiply all the numbers in it by 2. That still gives us a valid equation. The cost of 2 apples and 4 bananas would be 8 dollars.

2a + b = 5
2a + 4b = 8

Now, if it cost 5 dollars to get 2 apples and a banana, and 8 dollars to get 2 apples and 4 bananas, it stands to reason (assuming no discounts) that the difference in cost comes from the difference in goods. So we can subtract the first equation from the second, to get:

2a + b = 5
3b = 3

Again assuming no discounts, if 3 bananas cost $3, then one banana would cost $1.

2a + b = 5
b = 1

Finally, we can remove bananas from the equation and divide by 2 to find that an apple cost $2 last month.

2a = 4
b = 1

a = 2
b = 1

Notice that the only things we did were listed above - we multiplied (or divided) by numbers, and then we added (or subtracted) two things.

OK. Now let's do the calculation for this month.

2a + b = 5
a + 2b = 7

2a + b = 5
2a + 4b = 14

2a + b = 5
3b = 9

2a + b = 5
b = 3

2a = 2
b = 3

a = 1
b = 3

We have found that the new prices are $1 for an apple, and $3 for a banana. But one thing to notice - we in fact did the exact same calculations as we did before, just with different numbers on the right. This will be important later.

Hopefully I have already convinced you that we already use linear equations commonly. Next up, the process of solving generally, and some notation.
uncool is offline   topbottom
Old 05-10-2016, 03:06 PM   #2647954  /  #2
Heinz Hershold
Superior Member
 
Heinz Hershold's Avatar
 
Join Date: Sep 2011
Posts: 2,727
Heinz Hershold
Default

OK. That was easy. I have a feeling *easy* will not last much longer.

Bring it on!
__________________
RICHARD P. FEYNMAN
Some remarks on science, pseudoscience, and learning how to not fool yourself. Caltech’s 1974 commencement address.
If you’re doing an experiment, you should report everything that you think might make it invalid—not only what you think is right about it: other causes that could possibly explain your results.
Details that could throw doubt on your interpretation must be given, if you know them. You must do the best you can—if you know anything at all wrong, or possibly wrong—to explain it.
Heinz Hershold is offline   topbottom
Old 05-10-2016, 04:43 PM   #2648030  /  #3
Testycalibrated
incredibad
Age Of War Normal Champion
 
Testycalibrated's Avatar
 
Join Date: Feb 2010
Location: commensurate
Posts: 10,784
Testycalibrated
Default

Wow. Nice job.
__________________
I never met a man I didn't like.
-Will Rogers
Quote:
Sen. McCaskill: "It Is Unacceptable For Anybody To Refer To Hillary Clinton As A Corporate Whore"
Testycalibrated is offline   topbottom
Old 05-10-2016, 08:29 PM   #2648202  /  #4
Pingu
I did. F. Poste.
GLaDOS
 
Pingu's Avatar
 
Join Date: Mar 2008
Location: UK
Posts: 60,846
Pingu
Default

cool uncool
__________________
Join me at The Skeptical Zone
Pingu is offline   topbottom
Old 05-14-2016, 03:34 AM   #2649933  /  #5
MikeB
Senior Member
 
MikeB's Avatar
 
Join Date: Apr 2010
Location: So. Cal. USA
Posts: 322
MikeB
Default

This is a bit different from the solution path I was taught. I'm used to solving for one variable in terms of the other in one equation and substituting into the other equation.

So a + 2b = 4 resolves to a = 4-2b

Then substituting this for a into the other equation:

2(4-2b) +b =5 yields b=1

And so forth, same results...

ETA: Oh yeah, "Bite me Heinz" you so-called superior member! Heh.

Last edited by MikeB; 05-14-2016 at 03:38 AM.
MikeB is offline   topbottom
Old 05-14-2016, 07:49 AM   #2649935  /  #6
uncool
Senior Member
Plasmatron Champion
 
Join Date: Jul 2008
Posts: 15,106
uncool
Default

Quote:
Originally Posted by MikeB View Post
This is a bit different from the solution path I was taught. I'm used to solving for one variable in terms of the other in one equation and substituting into the other equation.

So a + 2b = 4 resolves to a = 4-2b

Then substituting this for a into the other equation:

2(4-2b) +b =5 yields b=1

And so forth, same results...

ETA: Oh yeah, "Bite me Heinz" you so-called superior member! Heh.
Yup, both good ways to do it - and both are widely taught. Linear algebra does it a slightly different way - in fact, it standardizes it, and that's what the next post will be.
uncool is offline   topbottom
Old 05-17-2016, 08:25 PM   #2650804  /  #7
Pandora
Betty Beep Boop
 
Pandora's Avatar
 
Join Date: Oct 2008
Posts: 14,480
Pandora
Default

Quote:
Originally Posted by MikeB View Post
This is a bit different from the solution path I was taught. I'm used to solving for one variable in terms of the other in one equation and substituting into the other equation.

So a + 2b = 4 resolves to a = 4-2b

Then substituting this for a into the other equation:

2(4-2b) +b =5 yields b=1

And so forth, same results...

ETA: Oh yeah, "Bite me Heinz" you so-called superior member! Heh.
My brain is insisting that your approach is the standard algebraic way to go about it - but it involves moving things from one side of the equation to the other. It involves rearranging things. Linear algebra doesn't let you rearrange things - you manipulate the entire formula, and you work across formulas rather than within.

Traditional algebra is intra-formula solving; Linear algebra is inter-formula solving
__________________
This is going to hurt me a lot more than it hurts you.
Pandora is offline   topbottom
Old 05-17-2016, 08:26 PM   #2650806  /  #8
Pandora
Betty Beep Boop
 
Pandora's Avatar
 
Join Date: Oct 2008
Posts: 14,480
Pandora
Default

Are you going to explain eigenvalues so they make sense? Those never made sense to me. That was a case of me memorizing the steps and plugging in the values.
__________________
This is going to hurt me a lot more than it hurts you.
Pandora is offline   topbottom
Old 05-18-2016, 12:49 PM   #2650988  /  #9
Brother Daniel
predisposition to antagonism
Kodos the Executioner Mod: HASH, Philo, Math, THC
 
Brother Daniel's Avatar
 
Join Date: Jun 2008
Location: on the shore of the North Atlantic
Posts: 15,670
Brother Daniel
Default

I don't want to derail uncool's thread when it has barely gotten started, so I'll just note that this
Quote:
Originally Posted by Pandora View Post
Linear algebra doesn't let you rearrange things
...is not true, and I'll bite my tongue regarding the rest of my objections.
__________________
Requiem: Everything we humans do is fully deserving of ridicule and mockery. Without a God to laugh, it falls to us enlightened fools to make up the difference.
Brother Daniel is offline   topbottom
Old 05-18-2016, 04:48 PM   #2651172  /  #10
Pandora
Betty Beep Boop
 
Pandora's Avatar
 
Join Date: Oct 2008
Posts: 14,480
Pandora
Default

Quote:
Originally Posted by Brother Daniel View Post
I don't want to derail uncool's thread when it has barely gotten started, so I'll just note that this
Quote:
Originally Posted by Pandora View Post
Linear algebra doesn't let you rearrange things
...is not true, and I'll bite my tongue regarding the rest of my objections.
We could get all sorts of techincal with that, but generally, teh shape of the formulas stay the same. If you start with ax+by=cz, you don't really shift to by = cz-ax. Sometimes, but the whole point of a matrix is kind of that you're not moving from one side of the equality to the other, you're manipulating the formulas as a whole entity in and of themselves.

Of course, it's also been a very long time since I did anything more than a pretty straightforward matrix solution. Hell, it's been about 7 years since last had to do anything more complicated than spell determinant. So there's a pretty good chance that I've brain dumped this bit!
__________________
This is going to hurt me a lot more than it hurts you.
Pandora is offline   topbottom
Old 05-18-2016, 04:54 PM   #2651183  /  #11
Brother Daniel
predisposition to antagonism
Kodos the Executioner Mod: HASH, Philo, Math, THC
 
Brother Daniel's Avatar
 
Join Date: Jun 2008
Location: on the shore of the North Atlantic
Posts: 15,670
Brother Daniel
Default

Generally, the scope of any given subfield of mathematics is defined by its subject matter, not by any artificial constraints on which techniques you're allowed to use.
__________________
Requiem: Everything we humans do is fully deserving of ridicule and mockery. Without a God to laugh, it falls to us enlightened fools to make up the difference.
Brother Daniel is offline   topbottom
Old 05-18-2016, 05:01 PM   #2651189  /  #12
Pandora
Betty Beep Boop
 
Pandora's Avatar
 
Join Date: Oct 2008
Posts: 14,480
Pandora
Default

Okay then. Whatever. Don't really care that much


ETA: Apparently I care a little bit. If you were pressed to make a general statement, how would you describe the difference between "regular" algebra and Linear Algebra?
__________________
This is going to hurt me a lot more than it hurts you.
Pandora is offline   topbottom
Old 05-18-2016, 05:53 PM   #2651214  /  #13
Brother Daniel
predisposition to antagonism
Kodos the Executioner Mod: HASH, Philo, Math, THC
 
Brother Daniel's Avatar
 
Join Date: Jun 2008
Location: on the shore of the North Atlantic
Posts: 15,670
Brother Daniel
Default

"Regular algebra" isn't a term I use.

Linear Algebra is about what you can do with vector spaces.
__________________
Requiem: Everything we humans do is fully deserving of ridicule and mockery. Without a God to laugh, it falls to us enlightened fools to make up the difference.
Brother Daniel is offline   topbottom
Old 05-18-2016, 06:10 PM   #2651221  /  #14
Pingu
I did. F. Poste.
GLaDOS
 
Pingu's Avatar
 
Join Date: Mar 2008
Location: UK
Posts: 60,846
Pingu
Default

Why "linear"?
__________________
Join me at The Skeptical Zone
Pingu is offline   topbottom
Old 05-18-2016, 06:49 PM   #2651236  /  #15
Pandora
Betty Beep Boop
 
Pandora's Avatar
 
Join Date: Oct 2008
Posts: 14,480
Pandora
Default

Quote:
Originally Posted by Brother Daniel View Post
"Regular algebra" isn't a term I use.
Couldn't come up with a better term. When I took the class, it was just "Algebra". But that didn't seem sufficient to distinguish it from Linear algebra, so I winged it.

Quote:
Originally Posted by Brother Daniel View Post
Linear Algebra is about what you can do with vector spaces.
And that means what? How are they different?
__________________
This is going to hurt me a lot more than it hurts you.
Pandora is offline   topbottom
Old 05-18-2016, 06:55 PM   #2651239  /  #16
Pingu
I did. F. Poste.
GLaDOS
 
Pingu's Avatar
 
Join Date: Mar 2008
Location: UK
Posts: 60,846
Pingu
Default

Well, the variables in linear algebra can be vectors and matrices. So you can solve y=bx, where y and x are matrices.

amirite?
__________________
Join me at The Skeptical Zone
Pingu is offline   topbottom
Old 05-18-2016, 07:07 PM   #2651244  /  #17
Mike PSS
Senior Member
 
Join Date: Mar 2008
Location: Florida
Posts: 11,478
Mike PSS
Default

Mike PSS is offline   topbottom
Old 05-18-2016, 08:28 PM   #2651304  /  #18
Brother Daniel
predisposition to antagonism
Kodos the Executioner Mod: HASH, Philo, Math, THC
 
Brother Daniel's Avatar
 
Join Date: Jun 2008
Location: on the shore of the North Atlantic
Posts: 15,670
Brother Daniel
Default

Quote:
Originally Posted by Pingu View Post
Why "linear"?
Good (and well timed) question.

My description was perhaps too broad, since there are things you can do with vector spaces that go outside the scope of Linear Algebra.

But suppose the only functions of vectors that we consider are those having the properties
f(u+v) = f(u) + f(v) for any vectors u and v;
f(cv) = cf(v) for any vector v and any scalar c.
(This is what we call a "linear function".)

That's a rather restricted set of functions, but it turns out that there's a tremendous amount of stuff to learn about them, so they're worthy of having entire courses dedicated to them. That's Linear Algebra.
__________________
Requiem: Everything we humans do is fully deserving of ridicule and mockery. Without a God to laugh, it falls to us enlightened fools to make up the difference.
Brother Daniel is offline   topbottom
Old 05-18-2016, 08:33 PM   #2651313  /  #19
Brother Daniel
predisposition to antagonism
Kodos the Executioner Mod: HASH, Philo, Math, THC
 
Brother Daniel's Avatar
 
Join Date: Jun 2008
Location: on the shore of the North Atlantic
Posts: 15,670
Brother Daniel
Default

Quote:
Originally Posted by Pandora View Post
And that means what?
That's what this whole thread will be about. uncool has plans for how to present it, and I expect xe will do a better job than I would.
__________________
Requiem: Everything we humans do is fully deserving of ridicule and mockery. Without a God to laugh, it falls to us enlightened fools to make up the difference.
Brother Daniel is offline   topbottom
Old 05-18-2016, 08:46 PM   #2651334  /  #20
Pingu
I did. F. Poste.
GLaDOS
 
Pingu's Avatar
 
Join Date: Mar 2008
Location: UK
Posts: 60,846
Pingu
Default

Quote:
Originally Posted by Brother Daniel View Post
Quote:
Originally Posted by Pingu View Post
Why "linear"?
Good (and well timed) question.

My description was perhaps too broad, since there are things you can do with vector spaces that go outside the scope of Linear Algebra.

But suppose the only functions of vectors that we consider are those having the properties
f(u+v) = f(u) + f(v) for any vectors u and v;
f(cv) = cf(v) for any vector v and any scalar c.
(This is what we call a "linear function".)

That's a rather restricted set of functions, but it turns out that there's a tremendous amount of stuff to learn about them, so they're worthy of having entire courses dedicated to them. That's Linear Algebra.
OK, thanks
__________________
Join me at The Skeptical Zone
Pingu is offline   topbottom
Old 05-18-2016, 09:59 PM   #2651405  /  #21
uncool
Senior Member
Plasmatron Champion
 
Join Date: Jul 2008
Posts: 15,106
uncool
Default

Quote:
Originally Posted by Pandora View Post
Okay then. Whatever. Don't really care that much


ETA: Apparently I care a little bit. If you were pressed to make a general statement, how would you describe the difference between "regular" algebra and Linear Algebra?
I'd say that linear algebra is the study of what you can do with just the two operations named before: addition and scalar multiplication. "Regular" algebra, to me, would include linear algebra - but also allow more operations.
uncool is offline   topbottom
Old 05-20-2016, 06:53 PM   #2652540  /  #22
Pandora
Betty Beep Boop
 
Pandora's Avatar
 
Join Date: Oct 2008
Posts: 14,480
Pandora
Default

Quote:
Originally Posted by Brother Daniel View Post
Quote:
Originally Posted by Pandora View Post
And that means what?
That's what this whole thread will be about. uncool has plans for how to present it, and I expect xe will do a better job than I would.
Fair enough
__________________
This is going to hurt me a lot more than it hurts you.
Pandora is offline   topbottom
Old 05-20-2016, 11:34 PM   #2652641  /  #23
el jefe
Senior Member
TR Pundit
 
el jefe's Avatar
 
Join Date: Oct 2008
Posts: 30,945
el jefe
Default

Quote:
Originally Posted by Pingu View Post
Quote:
Originally Posted by Brother Daniel View Post
Quote:
Originally Posted by Pingu View Post
Why "linear"?
Good (and well timed) question.

My description was perhaps too broad, since there are things you can do with vector spaces that go outside the scope of Linear Algebra.

But suppose the only functions of vectors that we consider are those having the properties
f(u+v) = f(u) + f(v) for any vectors u and v;
f(cv) = cf(v) for any vector v and any scalar c.
(This is what we call a "linear function".)

That's a rather restricted set of functions, but it turns out that there's a tremendous amount of stuff to learn about them, so they're worthy of having entire courses dedicated to them. That's Linear Algebra.
OK, thanks
one might add that linearity turns out to be an enormously useful property for simplifying and facilitating calculations and problem-solving in all kinds of contexts. furthermore, linear approximations are a standard and indispensible tool for studying an even broader class of problems. that's why so much studying was done on it in the first place, and hence why there is so much to learn about linear functions/equations/etc. people have developed a lot of mathematical machinery to take full advantage of linearity, everywhere it comes up.

and to go further on why "linear", one can think of the linearity property BD defined as an abstraction of the elementary equation for a line: y = mx + b. in fact, I suspect (though don't know) that is the origin of the term.

Last edited by el jefe; 05-20-2016 at 11:37 PM.
el jefe is offline   topbottom
Old 05-21-2016, 07:19 AM   #2652677  /  #24
Pingu
I did. F. Poste.
GLaDOS
 
Pingu's Avatar
 
Join Date: Mar 2008
Location: UK
Posts: 60,846
Pingu
Default

Well solving linear (or linearized) equations is mostly what I use it for
__________________
Join me at The Skeptical Zone
Pingu is offline   topbottom
 

  TalkRational Archive > Discussion > Mathematics

Tags
uncool's cool

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts
BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Forum Jump


All times are GMT. The time now is 04:47 AM.


Powered by vBulletin® Version 3.8.6
Copyright ©2000 - 2017, Jelsoft Enterprises Ltd.
Copyright ©2008 - 2017, TalkRational.org