Linear combinations and span | Vectors and spaces | Linear Algebra | Khan Academy

[@jennahaines9477]

I have been watching Khan since I was in middle school for math help. Now, I'm at Georgia Tech and STILL getting A's and B's in math because of this man. All my gratitude. You deserve highest honors for teaching the world.

[@dabidmydarling5398]

What was your high school GPA? I'm a junior and I'm planning to apply for Georgia tech.

[@jennahaines9477]

@@dabidmydarling5398 Oh gosh I can't remember. I know I was just shy of getting Zell Miller. So... I think 3.69? Had high ACT scores though and dual enrolled. I think the extra stuff helped since my GPA was just okay in terms of Tech's standards. You can always transfer in- if you really want to go there, you can definitely find a way to do that! it's a great school and I highly recommend it!

[@dabidmydarling5398]

@@jennahaines9477 Hi! I'm also dual enrolled. We have a similar GPA and I am also planning to take the ACT (hopefully I do good on it 🤞🤞) Thank you so much for giving your experience. This really gives me perspective on how hard close/far I am. Did you do any tech-related extracurriculars? Which extracurriculars should I be doing?

[@muhammadashshiddieqi8653]

ĹBBBŞČ MÅÝ F W Ù ÑĒĒĐ ĒŞP ČÒÙŘŞĒŞ Ñ ĶÀHVÈÌÑ ČÒFƏĔŞHÒP Ñ MZ.ŞHÌDĐÌĒQÌ ÇHÈÇĶÌŤĹBBBŞČ MÅÝ F W Ù ÑĒĒĐ ĒŞP ČÒÙŘŞĒŞ Ñ ĶÀHVÈÌÑ ČÒFƏĔŞHÒP Ñ MZ.ŞHÌDĐÌĒQÌ ÇHÈÇĶÌŤ

[@sanguinelinguine]

Georgia Tech can be really iffy. It’s probably more based on extracurricular activities. I had 4.0, 33 ACT and got in, while my friend had 4.0, 35 ACT and didn’t. I’d say if you have solid academics, really work on leadership and community service roles.

[@kyledomen2627]

This quote really helped me understand the concept. "If vector a and vector b spans R2, that tells us any vector in R2 can be represented by a linear combination of vector a and vector b"

[@mrjoni54]

3:47-3:52 " Hey Sal, why are you even introducing this idea of an linear combination?" "Beacause I wanna!"

[@adrianchivas2014]

Thank you khan for helping me understand this. I am taking Linear Algebra my sophomore year in college. Anyways, all I have to say is that you are the real MVP!!! Always, remember that you are changing lives and I am in college because of you my friend. On track to become an Electrical Engineer. On behalf of everyone that admires you, Thank you!!!

[@benpotter3278]

adrianchivas2014 I am In 7th grade and I'm learning this

[@jacobm7026]

by the way, i'm in kindergarten and am loving vector spaces and the alphabet

[@Dan-gc3ke]

smh I'm in pre-school and I'm learning this

[@ProLeopardx1]

@@katb9776 Nah, I don't know what school you went to but I wasn't taught vectors until calculus. I'm learning linear algebra right now after Calc 1, 2, and 3 (multivar). It's combined with multivar calc at my college and I hate it. Everyone's doing far worse in this than they did in calc 3.

[@SanaZub]

im in daycare and im learning this!!! just turned 3 yesterday! woot woot MATRICES R LITTTTTTTT

[@jappmayo1220]

I just paused this halfway to say THANK YOU for helping me to understand this!

[@Station1643]

but u asian, why u no understande ?

[@uddipanbaruah6021]

@@Station1643 abbeeyyy kije hbo mane

[@Station1643]

@@uddipanbaruah6021 abbe me no understand Indian

[@uddipanbaruah6021]

@@Station1643 its ok.. Jai Ai Axom

[@mahadevhatti5228]

@@Station1643 doe youe understande whate youe saide ??

[@nlnrose]

... you explain this better than my college matrix algebra teacher... wow I actually am starting to understand this.

[@vanessang4216]

You are a true blessing to the world

[@economicswithozzy]

2024 maths is still same 😂😂😂

[@Got-it747]

Because reality is still the same

[@BAMEADManiyar]

​@@Got-it747true 🤞

[@laian4152]

Lll

[@WalidFeghali]

I passed Linear Algebra because of these videos. You're the best, Khan!

[@pinklady7184]

Nah no noo, he da best and *BESTEST!*

[@majed1911]

13 years later I hope you can answer this question What can I do to practice in the best way possible

[@WalidFeghali]

@@majed1911 The best way to study is to start small. Start with the easiest stuff, and learn it well. If you don't understanding something early on, basic things, talk to a tutor in class, don't go straight to the professor. The tutors are great because they were where you are just recently. 13 years go by fast!

[@EwoudCP]

Wow. This is really useful. I've been trying to make sense of the idea of the span, but I tend to see things other people don't. More frustratingly, I tend to not see things that other people do. Thanks for the video. I'll definitely keep your playlists in my bookmarks.

[@mabeklvista]

why is this 10 times better explained than in lectures? they go so fast, here i can learn on my own speed. Thank you Sal!

[@alechenderson8410]

shoutout to anyone else with a bad linear algebra prof

[@buildalifeworthliving4551]

Lmaooooooo 🎉

[@TheBlackZoe]

Using this "pseudo-proof" system: drawing the vectors as a visual method is such a great representation of the idea of vector space. Thank you. Greetings from Greece.

[@selingermann5599]

I have finally reached ENLIGHTENMENT! Thank you.

[@eleazararce5716]

About to take this class for a second times after 13 years of being out of school. Its should be fun and interesting this time around!! I will be utilizing these videos too!!

[@bobbypesch5220]

vibing in my dorm rn but had to give a shoutout to Sal, this man got me into college

[@Zurh1994]

They've explained me this for 3 consecutive years. Every year I fear this explanation because I can't picture it, I don't know why is it useful, and now it's really like I can see the truth hahaha. How can you explain everything so nicely in every single video and my teachers be so useless? XD

[@krose7406]

i gave up and just started watching the lectures DURING class!!

[@drahoo302]

15:46 You Sir, just wrote it perfectly :-)

[@waseem1173]

Now thats a real teacher. man i love you . you're a life saver

[@rowanfrankenbach3625]

7:57 the "SPAN" he said it with gusto lmao. sal is legendary

[@Fallen_Fly]

Thank you, linear algebra and differential equations has changed my opinion about math from completely in love to loathing it. Hopefully with your videos I can get my passion back.

[@sherajr]

I'm so glad I found this before my midterm!!

[@VSpooner]

this makes understanding the math so much better, rather than just showing the process you manage to show the process of understanding.

[@fadlmasri]

the fact that this vid is 11 years old and it still helped me is outrageous

[@theotydingco7987]

18:00 How do you get from 3c2 = x2 - 2x1 to c2 = 1/3(x2 - x1) Where does the "2" in -2x1 go?

[@mandy1339]

+Theo Tydingco did you watch the video all the way to the end? you can see what happens

[@theotydingco7987]

+mandy1339 yeah figured it out.

[@patmull1]

+Theo Tydingco Yeah.... I was like "PHEW... thanks god" :D

[@sahil5185]

Yeah I was so stressed out until he corrected it lol!

[@fluffio2976]

*Comes back an hour after having paused the video* "The actual fuck has he drawn."

[@Batooya]

idk if you (the person teaching in the vid) will ever read this but I, and I'm sure others as well, have come to find you and your lessons to be really comforting

[@danielcohen9637]

How do you know so much? You know math, accounting,business, computers,physics and more.

[@SochoChun]

sweet... it's much better than my professor's lessons, this is a really good revision lesson before my test

[@SporkyG]

he corrects it at 19:26 :)

[@gibbytravis]

@18:22 how did you go from 3c2 = x2 - 2x1 to c2 = 1/3(x2 - x1)? Is that a mistake? Did you mean to say c2 = 1/3(x2 - 2x1)?

[@gibbytravis]

Never mind. I see that you corrected it.

[@nathandrouillard2927]

Thank you. These concepts aren't even that hard, but my prof speaks in VERY broken English and teaches very poorly on top of that so I absorb close to nothing in my lectures. These videos are a godsend

[@witnessforchrist7778]

Not even 3 minutes into it, I've learned more from Khan Academy so far than 8 weeks of linear algebra in class. My teacher sucks at explaining, he's all over the place, he has a strong accent, his hand writing sucks, nor does he explain terminology.

[@wjestick]

Magic formula. Watch this video before you cover LA in the lectures. Then use the lectures as revision! It is a lot easier to pay attention when you are actively listening. And with two reviews you are much more likely to retain the material.

[@roxymang1]

so helpful! i was stuck on a question all day reading the textbook, then i watched this video and now it's clear :D

[@danteeep]

exactly what happened to me

[@omardanial7351]

Thanks Sir Salman Khan.. You saved my semester of engineering.

[@McDreamOn]

Have an exam tomorrow over this stuff and I just had an ah ha! moment!!!!! So much clearer than my professor!

[@Ben1220]

You're absolutely right, it doesn't matter how many vectors you have, linear combinations are defined for any set of vectors. He probably should have chosen a different name for the first n, but this doesn't matter if you realise that the two n's are different. Well done for spotting that :P

[@hai2410]

@Ben1220 and @thomapple I think he's right actually. R^n is n-tuples of real numbers. If you're working in R^2, you have two dimensions to work in and so you get [x1,x2], R^3 [x1,x2,x3], so it follows from that.

[@river.]

13:11 finally understood span

[@haro123ish]

love you man, i always wondering what a span was. my teachers have failed to explain this so far. thank you

[@zhongqiren6794]

God bless you, Khan, Thank you very very very very much.

[@bgraaman89]

I might receive flak for saying this, but this Linear combination video of yours is much better than Dr. Gilbert Strang's MIT video. You've explained it much better than he did. Thank you.

[@Edwrath]

My University Day: -Sleep through morning lectures. -Watch khanacademy versions of my lectures -Play guitar all afternoon

[@adam.phelps]

@Ace12174 He multiplied by -2 so he could find one variable at a time - which in this case wasC2 - so after he found the value of C2 he could find the value of C1. Its called solving a system of linear equations by using elimination.

[@fliskmayhem]

@LostAngel01234 First of all, you'll need at least 3 diffrent vectors (call them v_1, v_2 and v_3). Each with dimension 3, so they'll look like (x,y,z). Second, it must be impossible to write one of these vectors as a linear combination of the other 2 vectors. I.e. : c_1 v_1 + c_2 v_2 + c_3 v_3 = (0,0,0) only if c_1 = c_2 = c_3 = 0, the three vectors span R3. If there's another solution for the c_i's, it means the vectors span up a plane or a straight line in R3, but not whole R3.

[@slykni8]

Wow, you are a real genius. You just cleared my doubt.❤❤❤

[@necatiakpnar5677]

your video uploaded in 2009 but still helping to people understand algebra. Thanks for everything!

[@danielgonzalezisaiev9643]

Love your videos! The thing that makes great lecture videos is the person talking and their pedagogic skills and you sir, YOU GOT EM'! Great presentation, great understandable language, thanks so much for your videos!

[@publicanimal]

You are a god. with the graphical representation of vector space R2 I finally understood the concept of linear combos. thanks

[@yusuf1597]

At 6:18 Sal turns into a Canadian.

[@dybiosol]

I see what you did there!

[@ataadi1460]

Everyone saw what he did there.Any issues?

[@evelynecheung1905]

2020 and this is still saving lives thank you so much !!

[@amayashenuri1147]

just an amazing teacher❣

[@desxse6309]

Thank you so much. After I watch this, I understand linear independent, linear dependent and why free variable relate to them.

[@zack_120]

4:50 - the graphic coordinates in increments of 2, not 1, is counterintuitive, reducing lecture efficiency.

[@razberrycreme]

You are an amazing teacher. Thank you!

[@mediocremathtutorials2098]

All the linear combinations of two vectors (in R3) are all of the points on a family of planes, when tied to a single point it represents a single plane. This is because the parametric form of a point P on a plane is P= a +Sb +Tc. Where a is a known point on the plane, S and T are scalars, and b and c are vectors ON the plane.

[@mediocremathtutorials2098]

Edit: the vectors a and b must be non-parallel.

[@bryannamelenhorst]

thank the lord i found these video's. you are an amazing teacher!

[@imegatrone]

I Really Like The Video Understanding linear combinations and spans of vectors From Your

[@oluwatoba11]

I think I need to take some moment of pause sometimes in the exams too! Great guy!!!

[@tsaras2838]

your voice gives me hope.

[@truesoundwave]

4:00 Yep.

[@chiupaulpaul]

I'm so glad you made this video.

[@lilonepj]

Thanks for the videos. Please add practice problems for the linear algebra topics!! I can only find 4 practice sets for the vectors section.

[@khashya786786]

This is real charity work!

[@krishonweb]

I love you man! What an explanation

[@MrAntihumanism]

love this!!

[@allarforheisman]

thank you Khan ! i’m taking linear algebra in my junior year in high school! This video helped a lot!

[@shrutikasansaria8499]

The explanation is just awesome 🤩

[@Sekkonal]

At 17:59 he multiplies both sides by 1/3 to remove the coefficient from 3C sub-2, and on the right side of the equation he drops X sub-1's coefficient? Is that an error or am I just not seeing where that coefficient went?

[@Darlenei]

I wondered the same? @Khan Academy ?

[@edgbaston149]

Thank you so much Sir🙏

[@fauziahassan9938]

Hey sir thank u sooo much for this video .....

[@anandib5401]

2 minutes silence for those who are watching this in 2021.

[@anandib5401]

By the way, I was just 1 y/o in 2009

[@brightknight0623]

@@anandib5401who

[@JesusAxolotl]

@Feroyn totally agree!

[@shibsankar4569]

Best and simple to understand

[@towbsmedia]

Cheers brother; great job. thank you

[@Robertlavigne1]

I cant tell you how much I love this man!! I want to kiss him on the mouth!!! 4 days till my LA final and I was getting so frustrated. Now its all clear!! Thank you so much!!!

[@Menor55672]

Thank you Mr. Khan

[@talha6k]

You are great! God bless....

[@sophmcamp]

binging many of these videos before my final because I rarely understood my professor’s explanations

[@Zurh1994]

Nobel prize for this guy, please!

[@MisterTutor2010]

"All linear combinations are linear combination" Furthermore all Tautologies are Tautological :)

[@carolllj]

thank u so much my lifesaver

[@Ace12174]

@GMmarine Oh, thanks, I was better at substitution compared to elimination so that's probably why I didn't understand

[@jhl849]

Thank you for very good explanation.

[@alikoteich6639]

Your crosshair is amazing for csgo

[@GiggleJuiceee]

I was 7 in 2009

[@danteeep]

I can't thank you enough for this video

[@azfanification]

you rock my khan!!

[@nonyvip1]

شكررا جزيلااا

[@dansam5467]

Thanks for the help

[@MrGreenCheddar]

A diagram drawn well will always make sense without requiring a single word. This 7:43 is not one of them.

[@rohitbhosle6521]

all bad teachers at least watch this videos because if they watch then students will love to learn things

[@ceciliaosorio7746]

Thank you so much for this very helpful video!

[@Ace12174]

This is definitely helping me understand span. It's starting to sink in! lol Thanks! There's one question I have though, why do you multiply (at 17:24) when finding any C1 and C2 by -2? thanks =) (Grade 12 Calculus and Vectors)

[@Earphony]

OOOOOOOOOOOOOOOOOOOOOOOOOOO now i get it. Thx Sal.

[@doaard9174]

شكرا

[@Sekkonal]

@Sekkonal And then I continued on he corrected it. :| That really needs to be annotated, because I was so confused that I had to double check my own methods.

[@elinorwang2705]

It was so helpful:) Thanks a lot!