+ V is a subspace (0-6:00) - V contains the 0 vector: (2:26-2:44) - Closed under multiplication: (2:47-4:23) - Closed under addition : (4:28-5:35) Subspace 3 points ; subspace & closure under (A & M) Examples using concepts: proving a subset (S) is/not a subspace - (08:55-14:17) Span (v1, v2, v3) is a valid subspace of R^n? - (14:21-19:42) Visualizing if a simple vector can be a subpsace. U = span([1, 1]) (19:43)
Amazing..I spent past 7 hours trying to understand subspaces from my notes with no success and now its crystal clear in just 23.5 minutes. A huge THANKS to You!!
This is so good. Teachers usually go straight into complex examples and totally roll over the concepts. You make the concepts clear and concise. Thanks so much!
In addition to that, he is great at explaining all these concepts to people who have little or no idea about the subject. Most people don't realize this, but this is extremely difficult to do for most people, but he is able to get into the mindset of a person who is new to this stuff, and explain it in such a way despite his great understanding of the subject. Thank you Sal for all your hard work..
If you are just learning mathematics in your college, so you don't need to go. We should not be in college just for a degree paper, do more things(-like starting a new projects,etc.)...
@@rahulkiroriwal8779 I have my degree in Mechanical Engineering and am working as a project engineer for construction projects and have started a business on the side ! Thanks for checking :D
Wow..... I spent five hours trying to learn this, and still could not comprehend the material. 20 minutes of this video, and everything makes sense now. This is awesome!!!
thanks! my lecturer is terrible at explaining this stuff and uses all the extremely abstract notation stuff that no body understands. your explanations are easy to understand and make difficult to grasp concepts simple. i love it!
I love his style and voice. His detailed persistent explanations show how much he is capable of relating to his audience and how much experience he has in teaching. I found him the best explainer that is out there for concepts that are typically confusing or need further breakdown.
Very cool explanation. I've been trying to teach myself this sort of stuff from books and Wikipedia. They're too dry for my current level of knowledge. Your dynamic presentation is exactly what I need and I'm sure will help me to understand other resources.
Thank you very much, my professor is blowing through this information leaving me wondering what just happened, you are helping me keep pace with the lessons
Thank you so much. When you stated between 5:10-5:30 that we're simply choosing vectors within the given parameters of our space V and then adding them together to see if the resultant vector was still in space V, everything finally clicked. I immediately understood what "closed under addition" and "closed under scalar multiplication" meant. Words cannot describe my gratitude! Thank you!
Thank you so much for taking the time to make this video! I really appreciate the fact that you take the effort to try to connect these concepts in many different ways, like in terms of things we've already learned, or what works and what doesn't, instead of just the formal mathematical definition. I mean, appreciate the formal mathematical definition, (I like math because it seeks "truth without error") but it's not much use if I don't understand things in the first place. Anyway, thanks!
Wow, thank you SOO much! My teacher did a HORRIBLE job explaining this. I've tried searching everywhere on youtube... Thanks man! Seeing it visually helps a lot more than just seeing the theorems or whatever proven with random variables with no significance.
Hey Sal, I looooove your videos, first of all. After taking Algebra 1 I told myself I would spend one summer learning all through Calc 1, and, thanks to you, this summer I am attending SacState as a Freshman in high school. I know that this topic is unrelated to the video and that you probably won't even have time to read this, but I was wondering if you have any intuition that you could teach me regarding visualization of Lie groups and symmetries because it's really difficult for me to imagine. I understand how group theory works, but imagining these things as circles and large manifolds is a giant leap from just sets of numbers. Thanks a ton for your videos; I don't know what I would have done without you.
Nice, I did calc 3 and Diff Eq in 8th grade. I also attended USAMO. Are you a full time student at the university, or are you just taking some courses there? I don't think you can graduate high school right in just one summer. Also, I'd imagine that there are gaps in your mathematical knowledge because you haven't done enough geometry, and you did not spend enough time on the prerequisites of calc. Additionally, doing higher math does not mean a lot. I did Calc BC in school as a 7th grader, and I don't consider my 7th grade self all that smart. Doing calc does not require as much creative thinking as math competitions. Try the first competition to the IMO, which is the AMC 10/12. Try some problems there and see how you do. This competition will probably be hard for someone without much experience. Next, there will be the AIME, which is much much harder. If you want to be better at math, I suggest participating in these competitions.
Congratulations man !!! They way you teach is very clear-easy-good,not only because you explain them with great way,but because you used examples,and you used Cartesian plane to show how vectors works in reality. That really helps,because to understand something in maths,you need to implement it. Keep the good job !!!
dude, Partrick is good and khan is good, they r different in their styles. khan is the very best in explaining the derivation and the origin of things, partick is gr8 in solving examples.
After watching this video I just have the urge to show this to my Linear Algebra lecturer and ask him, why his explanations are not this simple to understand?????????????????????????????????????????????????????????
@123aellis in example (x1 in R and x1>0, x2 in R) this will be a subspace if any c that is in real set will satisfy above conditions. Your 'c' does satisfy condition -> v[2,4]*3=[6,12]. 6 is greater than 0. However -3 will product vector [-6,-12]. -6 does not satisfy condition x1>0. Important is that it must be valid for any c's in R.
Salman Khan, the creator of Khan Academy, holds a bachelors degree in mathematics, a bachelors degree in electrical engineering and computer science, as well as a masters degree in electrical engineering and computer science from MIT, and an MBA from Harvard Business School. He is unbelievably talented, so instead of complaining you should be grateful for this amazing resource that you can implement in your education.
You explained to me in under half an hour wait University professor can't explain in 3 lectures. Thank you so much, I understood a hole section in this video
A scalar is any number contained within the real numbers, which includes zero and every single positive and negative number. That also includes all of the fractions and decimals.
Question: Aren't the tests for closure for both scalar mult. and vector add. arbitrary? If you used c=3 for scalar multiplication, you would have had a new vector, still with x>0, and would answer that it is closed, and therefore is a subspace...? I'm having trouble getting my head around the idea of this...
I think of it as a subset being an open community of vectors whereas a subspace is a closed community of vectors. Since you can have a subset of Rn without it being closed under multiplication or addition or having the zero vector since it can span some or all of Rn. However, to get a subspace, you have to have it be in a subset that satisfies having the zero vector, closed under addition and multiplication.
would it then be correct to say that: Any linear combinations of vectors in a subspace must be equal to a vector that is also in the subspace ? kind of like the closure under addition? just with linear combinations?
In summary, In order to define subspace using vector we need to define for R^n i.e. for all real value, exculding a certain ranges would bring up the possibilty of subset instead of subspace, SPAN by default are defined for R, hence they automatically fall into subspace cateogory. If there is any example that doesn't include R i.e real domain, please do provide. I think one you have mentioned would be n dimension zero vector, the addition or linear combination again results in zero vector, which is exception of trend, but for non-zero n-dimensional vector space to be termed as subspace, it needs to be defined for 'R' . Note: Here R represents real domain.
I won't say where I'm attending school, but geez, can you come teach here, I'm scrwed with these gtfs, well I have you so guess not, thanks brother, hope you get this,
For the example of whether U=span{v_1, v_2, v_3} is a subspace of R^n, don't v_1, v_2, and v_3 also have to have n components, as in they are a subset of R^n?
is a point(that is not equal to zero) also a subspace? If we think of it as a position vector, it actually passes through the origin. that's why I'm curious
Thank you veeeeeeeeeeeeery muuch!! Come to our university and teach it us, intead of our prof :)! I will recommend it to everyone! GREAT! SIMPLY GREAT!
Trying to think in an abstract way about subspaces... could you say that a subspace is essentially an extension of the concept of an infinite line (1 dimensional) or plane (2 dimensional) into m dimensions, m
+ V is a subspace (0-6:00)
- V contains the 0 vector: (2:26-2:44)
- Closed under multiplication: (2:47-4:23)
- Closed under addition : (4:28-5:35)
Subspace 3 points ; subspace & closure under (A & M)
Examples using concepts:
proving a subset (S) is/not a subspace - (08:55-14:17)
Span (v1, v2, v3) is a valid subspace of R^n? - (14:21-19:42)
Visualizing if a simple vector can be a subpsace. U = span([1, 1]) (19:43)
+Tau Ceti The world needs more people like u
Thanks
Amazing..I spent past 7 hours trying to understand subspaces from my notes with no success and now its crystal clear in just 23.5 minutes. A huge THANKS to You!!
This is so good. Teachers usually go straight into complex examples and totally roll over the concepts. You make the concepts clear and concise. Thanks so much!
me looking in my linear algebra book: dafuq???
me looking in the same book after a video from khanacademy: Aha!
facts
init
fact
factos
In addition to that, he is great at explaining all these concepts to people who have little or no idea about the subject. Most people don't realize this, but this is extremely difficult to do for most people, but he is able to get into the mindset of a person who is new to this stuff, and explain it in such a way despite his great understanding of the subject. Thank you Sal for all your hard work..
hello there , how are u . its been 8 years
@@rahulkiroriwal8779 now 11
You are better than my professor lol. I dont know why I attend college.....
So you can get a real degree?
Yeah, to get degree so you can get a job, that's so annoying
@@That_One_Guy... I'm in college and the companies I'm aiming for don't need degrees. They're just insanely hard for everyone to get into
If you are just learning mathematics in your college, so you don't need to go. We should not be in college just for a degree paper, do more things(-like starting a new projects,etc.)...
ㅇㅇ
you accomplished in 23 mins what my linear algebra prof couldn't in two 2 hour lectures.
hey how u doing , its been 8 years ?
@@rahulkiroriwal8779 I have my degree in Mechanical Engineering and am working as a project engineer for construction projects and have started a business on the side ! Thanks for checking :D
@@lolalukie713 wow really great keep growing
@@lolalukie713 how about now , you good ?
Wow..... I spent five hours trying to learn this, and still could not comprehend the material. 20 minutes of this video, and everything makes sense now. This is awesome!!!
thanks! my lecturer is terrible at explaining this stuff and uses all the extremely abstract notation stuff that no body understands. your explanations are easy to understand and make difficult to grasp concepts simple. i love it!
I love his style and voice. His detailed persistent explanations show how much he is capable of relating to his audience and how much experience he has in teaching. I found him the best explainer that is out there for concepts that are typically confusing or need further breakdown.
Very cool explanation. I've been trying to teach myself this sort of stuff from books and Wikipedia. They're too dry for my current level of knowledge. Your dynamic presentation is exactly what I need and I'm sure will help me to understand other resources.
The way you define everything in simplistic terms is amazing. Thank you so much.
superb voice superb explanation superb colors why we dont have teachers like u damn
You deserve a noble price. Thanks
Nobel Prize? :P
Nolen Lah Nope, just a noble one
inteusproductions :DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
inteusproductions Haha, he sure does, but none are given for teaching.
Thank you sir for making it 100% easier. My professor sucks af.
Thank you very much, my professor is blowing through this information leaving me wondering what just happened, you are helping me keep pace with the lessons
Thank you sal, you are a great example of a good human being. With all negativity around the world you give me hope. Thank you so so much :)
Thank you so much. When you stated between 5:10-5:30 that we're simply choosing vectors within the given parameters of our space V and then adding them together to see if the resultant vector was still in space V, everything finally clicked. I immediately understood what "closed under addition" and "closed under scalar multiplication" meant. Words cannot describe my gratitude! Thank you!
I personally am thankful for your service sir, thank you very much.
Haha if you look at the statistics. The viewer count always goes up before exam periods :P
Im a student of statistics and i can vouch for that
crazy thought you got there, I wonder why that is...
how you figured that out genius?
I LOVE YOU. This helped so much. was freaking out about this class and was about to drop it, but now i understand enough to not panic.
Thank you so much for taking the time to make this video! I really appreciate the fact that you take the effort to try to connect these concepts in many different ways, like in terms of things we've already learned, or what works and what doesn't, instead of just the formal mathematical definition. I mean, appreciate the formal mathematical definition, (I like math because it seeks "truth without error") but it's not much use if I don't understand things in the first place. Anyway, thanks!
Wow, thank you SOO much! My teacher did a HORRIBLE job explaining this. I've tried searching everywhere on youtube... Thanks man! Seeing it visually helps a lot more than just seeing the theorems or whatever proven with random variables with no significance.
Hey Sal,
I looooove your videos, first of all. After taking Algebra 1 I told myself I would spend one summer learning all through Calc 1, and, thanks to you, this summer I am attending SacState as a Freshman in high school. I know that this topic is unrelated to the video and that you probably won't even have time to read this, but I was wondering if you have any intuition that you could teach me regarding visualization of Lie groups and symmetries because it's really difficult for me to imagine. I understand how group theory works, but imagining these things as circles and large manifolds is a giant leap from just sets of numbers. Thanks a ton for your videos; I don't know what I would have done without you.
Nice, I did calc 3 and Diff Eq in 8th grade. I also attended USAMO.
Are you a full time student at the university, or are you just taking some courses there? I don't think you can graduate high school right in just one summer. Also, I'd imagine that there are gaps in your mathematical knowledge because you haven't done enough geometry, and you did not spend enough time on the prerequisites of calc.
Additionally, doing higher math does not mean a lot. I did Calc BC in school as a 7th grader, and I don't consider my 7th grade self all that smart. Doing calc does not require as much creative thinking as math competitions.
Try the first competition to the IMO, which is the AMC 10/12. Try some problems there and see how you do. This competition will probably be hard for someone without much experience. Next, there will be the AIME, which is much much harder. If you want to be better at math, I suggest participating in these competitions.
Congratulations man !!!
They way you teach is very clear-easy-good,not only because you explain them with great way,but because you used examples,and you used Cartesian plane to show how vectors works in reality. That really helps,because to understand something in maths,you need to implement it. Keep the good job !!!
You are a better teacher than my professor. Thanks for putting the video up
dude, Partrick is good and khan is good, they r different in their styles. khan is the very best in explaining the derivation and the origin of things, partick is gr8 in solving examples.
This is great. I just discovered your library as well. Great stuff!!
crazy to think i'm paying £9k a year tuition fees, but learn the course from these videos! Thankyou! :)
Linear algebra test tomorrow not worried about this topic thanks to you!!!! Great vid!!
After watching this video I just have the urge to show this to my Linear Algebra lecturer and ask him, why his explanations are not this simple to understand?????????????????????????????????????????????????????????
This guy is just unbelievale !! Thumbs upp!!!
@123aellis in example (x1 in R and x1>0, x2 in R) this will be a subspace if any c that is in real set will satisfy above conditions. Your 'c' does satisfy condition -> v[2,4]*3=[6,12]. 6 is greater than 0. However -3 will product vector [-6,-12]. -6 does not satisfy condition x1>0. Important is that it must be valid for any c's in R.
wow this is awesome. this is SO much more helpful than any lecture I've been to this year!
These videos are teaching me more than my professor does. You may be saving my grade.
Salman Khan, the creator of Khan Academy, holds a bachelors degree in mathematics, a bachelors degree in electrical engineering and computer science, as well as a masters degree in electrical engineering and computer science from MIT, and an MBA from Harvard Business School. He is unbelievably talented, so instead of complaining you should be grateful for this amazing resource that you can implement in your education.
You are truly a great teacher! Than you for the posting!
More gratitudes should be granted to you Sir
Seriously this explanation was good im so happy i typed it in youtube... defo liked and subbed
I'm learning crazy much! You already feel like my best friend dear khan academy mystery man.
You taught me in 25 minutes what my professor took 2 hours to cover in class. Thank you!
it is not the first time you have been facing like that a new subject, so that you understood in 25 min.
You explained to me in under half an hour wait University professor can't explain in 3 lectures. Thank you so much, I understood a hole section in this video
CLEARLY!
thank you sir. u should receive like a humanitarian award for the number of times u have saved my life lol
I highly recommend this video. The presence of a graph makes it much easier to comprehend!
A scalar is any number contained within the real numbers, which includes zero and every single positive and negative number. That also includes all of the fractions and decimals.
Thanks to sal i now understand everything about basis, vector spaces and every other thing right before my tomorrow's exam... Thanks sal
You're presentations are very clear and you're a great teacher as well.Thank you very much for these videos!
it really saved me time for understanding these materials, thanks a lot.
wow what an amazing explanation! I spent too much time looking in notes and couldn't get much out of it
If I have Khanacademy on youtube, what is the point of staying in my math class? Well, I have more time to study other classes! Yay!
this was such a great review before my first test
Making up where my prof lacks - every time. Thanks for your useful explanations and videos.
Question:
Aren't the tests for closure for both scalar mult. and vector add. arbitrary? If you used c=3 for scalar multiplication, you would have had a new vector, still with x>0, and would answer that it is closed, and therefore is a subspace...? I'm having trouble getting my head around the idea of this...
Thanks you VERY MUCH for the nice, simple, useful explanation. God bless you
You are a life saver, thank you! I fully understand it now.
thanks. you took 30 mins to to untwine the confusion my professor infused in me for one semester
I think of it as a subset being an open community of vectors whereas a subspace is a closed community of vectors.
Since you can have a subset of Rn without it being closed under multiplication or addition or having the zero vector since it can span some or all of Rn. However, to get a subspace, you have to have it be in a subset that satisfies having the zero vector, closed under addition and multiplication.
Best tutorials on linear algebra!
Well, this made more sense than my teachers lecture. Thank you!
You are great. Now this concept is clear crystal for me.
thanks so much, you help me learn all these math concepts that I were unable to grasp in class when I was in school
would it then be correct to say that: Any linear combinations of vectors in a subspace must be equal to a vector that is also in the subspace ?
kind of like the closure under addition? just with linear combinations?
Single-handedly saving my life
I have a quiz tomorrow God bless you sal!! 😭
Perfect! Just what i needed! Thx alot
Thanks a ton for the video, hopefully i'll pass my final because of this lol
THanks! It makes so much sense now.
And you are officially my new teacher
In summary, In order to define subspace using vector we need to define for R^n i.e. for all real value, exculding a certain ranges would bring up the possibilty of subset instead of subspace, SPAN by default are defined for R, hence they automatically fall into subspace cateogory. If there is any example that doesn't include R i.e real domain, please do provide. I think one you have mentioned would be n dimension zero vector, the addition or linear combination again results in zero vector, which is exception of trend, but for non-zero n-dimensional vector space to be termed as subspace, it needs to be defined for 'R' .
Note: Here R represents real domain.
u know, I wonder why i even go to my math classes anymore...
oh wait, potential attendance quiz and knowing what to look up on Khan
I won't say where I'm attending school, but geez, can you come teach here, I'm scrwed with these gtfs, well I have you so guess not, thanks brother, hope you get this,
8:42 - To meet the 3 conditions as being a subspace means the vector contains ALL 0s, which means useless, or not?
it really helped me... thanku soo much...keep uploading more vedios...
thanks buddy you made my day
once again, you have saved me. thank you sir!
For the example of whether U=span{v_1, v_2, v_3} is a subspace of R^n, don't v_1, v_2, and v_3 also have to have n components, as in they are a subset of R^n?
Other than that, your video helped clear everything up. Thank you!
I was completely lost before watching only 25 seconds on this video.
now I am a linear algebra prof
That’s pretty clear! Thank you
I feel ya. We're using that book here as well.
is a point(that is not equal to zero) also a subspace? If we think of it as a position vector, it actually passes through the origin. that's why I'm curious
What if you multiplied the vector x at 16:18 by -1 like you did at 13:04 would it not be closed under scalar multiplication, just like the other one?
Thank you veeeeeeeeeeeeery muuch!!
Come to our university and teach it us, intead of our prof :)!
I will recommend it to everyone! GREAT! SIMPLY GREAT!
Amazing, you teach 10 times better than my school does, god this world is weird.
thanks ALOT u saved me
i wish my teacher was like him
Thank you for these tutorials. They are very compact
Thank you so much!
my professor should spend the class time showing us khan's video..
Nice class!
Thanks, it really helps me a lot.
Trying to think in an abstract way about subspaces... could you say that a subspace is essentially an extension of the concept of an infinite line (1 dimensional) or plane (2 dimensional) into m dimensions, m
+LayZ K1D of course if m=n (not m
thanks for saving my life
you guys should be thankful that this kinda help is even available and its free.....
Just wanted to ask at 11:00 why vectors were selected in only 1st and 2nd quadrant??
Something just completely clicked. Thanks.
wah!!
superb video
GREAT IT IS.
this helped so much thank you soo much!
Sometimes you question yourself the usefulness of certain concepts.
u just got to get that answer right buddy
@@7kwm oh is that how it is 🤣