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

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Understanding linear combinations and spans of vectors
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Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
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[@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"

[@economicswithozzy]

2024 maths is still same 😂😂😂

[@Got-it747]

Because reality is still the same

[@mrjoni54]

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

[@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.

[@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

[@WalidFeghali]

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

[@pinklady7184]

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

[@vanessang4216]

You are a true blessing to the world

[@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.

[@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.

[@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!!

[@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!

[@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.

[@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!

[@desxse6309]

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

[@VSpooner]

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

[@bobbypesch5220]

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

[@selingermann5599]

I have finally reached ENLIGHTENMENT! Thank you.

[@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

[@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

[@kjz8923]

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

[@drahoo302]

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

[@alechenderson8410]

shoutout to anyone else with a bad linear algebra prof

[@buildalifeworthliving4551]

Lmaooooooo 🎉

[@haro123ish]

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

[@krose7406]

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

[@necatiakpnar5677]

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

[@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

[@waseem1173]

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

[@SoloPro]

your videos are actually enjoyable to watch... hope the best for you for being this kind to share your knowledge with us all :D Keep it up! PEACE

[@sherajr]

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

[@danielcohen9637]

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

[@zhongqiren6794]

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

[@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.

[@razberrycreme]

You are an amazing teacher. Thank you!

[@McDreamOn]

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

[@oluwatoba11]

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

[@rowanfrankenbach3625]

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

[@ceciliaosorio7746]

Thank you so much for this very helpful video!

[@SochoChun]

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

[@sophmcamp]

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

[@imegatrone]

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

[@publicanimal]

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

[@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!

[@bryannamelenhorst]

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

[@chiupaulpaul]

I'm so glad you made this video.

[@fadlmasri]

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

[@damnfreakingsien]

THANK YOU!!! i was finally able to visualise this frustrating concept. Time to rush my homework before deadline...... :(

[@fluffio2976]

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

[@elinorwang2705]

It was so helpful:) Thanks a lot!

[@SporkyG]

he corrects it at 19:26 :)

[@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)

[@shibsankar4569]

Best and simple to understand

[@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.

[@omardanial7351]

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

[@shrutikasansaria8499]

The explanation is just awesome 🤩

[@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.

[@jhl849]

Thank you for very good explanation.

[@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

[@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.

[@siavasheghlimi6945]

Cheers brother; great job. thank you

[@danteeep]

I can't thank you enough for this video

[@raymondkazibweAwesomenessjust]

glad you noticed that error !I was gobsmacked initially!

[@talha6k]

You are great! God bless....

[@krishonweb]

I love you man! What an explanation

[@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.

[@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

[@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!!!

[@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?

[@MrAntihumanism]

love this!!

[@ExecutiveGoth]

This saved me for my high school exams thanks so much

[@ieatnoodls]

I'm out of words how I can finally understand this

[@Nandeesh_N]

@Khan Academy Should the scalar value which is used to multiply the vector belongs to a real value? if yes, at the end, the c2 obtained is a fraction.

[@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.

[@zack_120]

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

[@PrabhuShreeRamKiJai]

hey Sal, Can you please tell me why 3A+(-2B) vector is parallel to Y axis like this you had shown ....?/

[@nonyvip1]

شكررا جزيلااا

[@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.

[@amayashenuri1147]

just an amazing teacher❣

[@organicstore8557]

Great job Sir .

[@evelynecheung1905]

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

[@fauziahassan9938]

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

[@JesusAxolotl]

@Feroyn totally agree!

[@tsaras2838]

your voice gives me hope.

[@GiggleJuiceee]

I was 7 in 2009

[@arnabpaul7215]

wonderful lecture

[@TheJebacina]

Could you show how maximum combination of 2 vectors can be done?

[@disornr]

God bless you!!

[@Menor55672]

Thank you Mr. Khan

[@doaard9174]

شكرا

[@river.]

13:11 finally understood span

[@Earphony]

OOOOOOOOOOOOOOOOOOOOOOOOOOO now i get it. Thx Sal.

[@ksb916]

Thx this really helps :)

[@khashya786786]

This is real charity work!

[@dansam5467]

Thanks for the help

[@alikoteich6639]

Your crosshair is amazing for csgo

[@Ace12174]

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

[@rohitbhosle6521]

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

[@carolllj]

thank u so much my lifesaver

[@fawziyahrahman1030]

Thank you