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 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!
@@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?
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.
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"
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!!!
@@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.
@@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!
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.
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.
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!!
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
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.
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
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
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.
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.
@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.
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!
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.
@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.
@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.
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
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.
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)
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!!!
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.
What was your high school GPA? I'm a junior and I'm planning to apply for Georgia tech.
@@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!
@@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?
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
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.
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"
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!!!
adrianchivas2014 I am In 7th grade and I'm learning this
by the way, i'm in kindergarten and am loving vector spaces and the alphabet
smh I'm in pre-school and I'm learning this
@@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.
im in daycare and im learning this!!! just turned 3 yesterday! woot woot MATRICES R LITTTTTTTT
I just paused this halfway to say THANK YOU for helping me to understand this!
but u asian, why u no understande ?
@@Station1643 abbeeyyy kije hbo mane
@@uddipanbaruah6021 abbe me no understand Indian
@@Station1643 its ok.. Jai Ai Axom
@@Station1643 doe youe understande whate youe saide ??
... you explain this better than my college matrix algebra teacher... wow I actually am starting to understand this.
3:47-3:52 " Hey Sal, why are you even introducing this idea of an linear combination?"
"Beacause I wanna!"
I passed Linear Algebra because of these videos. You're the best, Khan!
Nah no noo, he da best and *BESTEST!*
13 years later I hope you can answer this question
What can I do to practice in the best way possible
@@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!
You are a true blessing to the world
2024 maths is still same 😂😂😂
Because reality is still the same
@@Got-it747true 🤞
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.
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.
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!!
I have finally reached ENLIGHTENMENT!
Thank you.
shoutout to anyone else with a bad linear algebra prof
Lmaooooooo 🎉
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!
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
vibing in my dorm rn but had to give a shoutout to Sal, this man got me into college
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.
this makes understanding the math so much better, rather than just showing the process you manage to show the process of understanding.
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
so helpful! i was stuck on a question all day reading the textbook, then i watched this video and now it's clear :D
exactly what happened to me
How do you know so much? You know math, accounting,business, computers,physics and more.
Now thats a real teacher. man i love you . you're a life saver
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
your video uploaded in 2009 but still helping to people understand algebra. Thanks for everything!
15:46 You Sir, just wrote it perfectly :-)
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.
i gave up and just started watching the lectures DURING class!!
Thanks for the videos. Please add practice problems for the linear algebra topics!! I can only find 4 practice sets for the vectors section.
Wow, you are a real genius.
You just cleared my doubt.❤❤❤
the fact that this vid is 11 years old and it still helped me is outrageous
love you man, i always wondering what a span was. my teachers have failed to explain this so far. thank you
sweet... it's much better than my professor's lessons, this is a really good revision lesson before my test
he corrects it at 19:26 :)
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.
7:57 the "SPAN" he said it with gusto lmao. sal is legendary
Have an exam tomorrow over this stuff and I just had an ah ha! moment!!!!! So much clearer than my professor!
I'm so glad I found this before my midterm!!
You are a god. with the graphical representation of vector space R2 I finally understood the concept of linear combos. thanks
Thank you so much. After I watch this, I understand linear independent, linear dependent and why free variable relate to them.
2020 and this is still saving lives thank you so much !!
God bless you, Khan, Thank you very very very very much.
I Really Like The Video Understanding linear combinations and spans of vectors From Your
just an amazing teacher❣
18:00
How do you get from 3c2 = x2 - 2x1
to c2 = 1/3(x2 - x1)
Where does the "2" in -2x1 go?
+Theo Tydingco did you watch the video all the way to the end? you can see what happens
+mandy1339 yeah figured it out.
+Theo Tydingco Yeah.... I was like "PHEW... thanks god" :D
Yeah I was so stressed out until he corrected it lol!
@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.
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!
13:11 finally understood span
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.
your voice gives me hope.
Thanks Sir Salman Khan..
You saved my semester of engineering.
@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.
Cheers brother; great job. thank you
Hey sir thank u sooo much for this video .....
@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.
thank you Khan ! i’m taking linear algebra in my junior year in high school! This video helped a lot!
Thank you Mr. Khan
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
This saved me for my high school exams thanks so much
I'm so glad you made this video.
I love you man! What an explanation
binging many of these videos before my final because I rarely understood my professor’s explanations
Thanks for the help
thank the lord i found these video's. you are an amazing teacher!
THANK YOU!!! i was finally able to visualise this frustrating concept.
Time to rush my homework before deadline...... :(
Best and simple to understand
You are great! God bless....
thank u so much my lifesaver
At 6:18 Sal turns into a Canadian.
I see what you did there!
Everyone saw what he did there.Any issues?
*Comes back an hour after having paused the video* "The actual fuck has he drawn."
4:50 - the graphic coordinates in increments of 2, not 1, is counterintuitive, reducing lecture efficiency.
@Feroyn totally agree!
The explanation is just awesome 🤩
Thank you so much Sir🙏
Thank you 🙏
love this!!
شكررا جزيلااا
You are an amazing teacher. Thank you!
@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)?
Never mind. I see that you corrected it.
OOOOOOOOOOOOOOOOOOOOOOOOOOO now i get it.
Thx Sal.
I think I need to take some moment of pause sometimes in the exams too! Great guy!!!
Thank you for very good explanation.
Your crosshair is amazing for csgo
شكرا
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.
Edit: the vectors a and b must be non-parallel.
thank you sir
@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.
This is real charity work!
you rock my khan!!
I can't thank you enough for this video
Thank you so much. This completely cleared up all the material I'd learned in class.
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)
glad you noticed that error !I was gobsmacked initially!
4:00 Yep.
Great job Sir .
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!!!
God bless you!!
My University Day:
-Sleep through morning lectures.
-Watch khanacademy versions of my lectures
-Play guitar all afternoon
wonderful lecture
Thank you so much for this very helpful video!