This video introduces one of the most important ideas of linear algebra - subspaces. More information on subspaces can be found in Gilbert Strang's excellent course - see Linear Algebra lecture 5
This is amazing. Just walked out of a lecture at the University of Melbourne on subspaces, understood nothing. Within 2 minutes of your video, I learned more than a whole hour in the lecture. Thanks! Subscribed :D
I usually dislike the way people try to explain certain concepts on TH-cam. This one was simple, clear to understand, and you summed it up nicely. Thank you for the post, and I agree that more videos from you would be excellent for those who are struggling on the subject matter!
This is the first time I saw a graphical representation of what a subspace is. I wonder why they don't show this in universities? Anyway thanks for the help, i understand it now.
Joseph Evora I noticed the same thing. I think it's stupid, because it really did help me understand the concept better. And of course I do realise that it isn't that easy for 4 or more dimensions but it's certainly helpfull when you're just starting with linear algebra.
Wdf? I have gone through many undergrad and a quite a few grad level courses which required the knowledge of what a subspace is, this is the first time it has ever made sense. Thank you!
I LOVE this! I feel like everyone teaches linear algebra incorrectly, because everyone tries to teach definitions, never any graphs. But I feel like linear algebra should be taught more graphically like you just did, and it makes so much more sense this way. Thank you so much!
I really appreciate this. This video is a breath of fresh air after dealing with lectures that use too many abstractions that don't really get to the point at times.
DAMMIT! A day before the finals and I am finding all these awesome videos....my professor makes everything so damn complicated for not apparent reason! Thanks!
Thank you very much. I have my first midterm in linear algebra on Friday and the concept of subspaces was still a mystery to me. Your explanation was perfect and I finally understand it.
Thank you! I was lost because I couldn't understand what a subspace was. Just in a few minutes I finally understood something that's been haunting me for a while. I couldn't go to class the day the topic was introduced in Algebra to make things worse.
This is so awesome!!! This video has just made my day!!! Now I know what I need to about subspaces!! Thank you so much for producing and posting it!! :) :) :) :)
Do subspaces need to go on to infinity then (whether a line, plane...)? So that adding two vectors will always land inside the subspace and not go outside of it.
For the second example, isn't that also not closed under addition? All the vectors should start at the origin and end up on the line, and then if you add two vectors from that line, you do not end up on the line. Am I misunderstanding?
I have question. Is it correct to oversimplify this: Can i say that a space(or plane?) is basically just bunch of dots with coordinates (depending on the dimension) and one vector is two selected random dots with distance and orientation between those two? So a subspace is random dots with specified range(either in a line of dots or something else covering bigger area of dots) and if one or both dots end out of that specified range it's not a subspace?
The best way to imagine subspace is as an upside down inverted cone normal space is the widest part of the cone while the smallest part is deepest part of subspace, Not unlike a black hole or a wormhole visually it's not because you can move through subspace, coordinates at the widest part of the cone are further apart then the coordinates at the the smaller side making travel between two points far shorter in subspace depending on how deep into subspace you can go. You are not actually bending or compressing spacetime you are decompressing it around the object.
What do you mean (2 minutes) you talked about if there's no zero vector there, then there's a constant you can multiply by that will not be in the subspace? Not understanding what that means. Are you saying that if there's no zero vector then when you multiply by zero it will take you out of the subspace?
Laura W For any subspaces, zero vectors must be included regardless of whether the scalar is zero or not. If the zero vector is not in W then it is not a subspace.
Intuitively, a subspace would make sense to be defined as any space which is a subset of another space. But for some reason, mathematicians decided that a subspace also has to equal to the span of all vectors in the subspace.
In your 1st example of what "is" a subspace, you mention that you can add any two vectors and still be within the subspace of the line. How is this possible? and why must you only be confined to choosing vectors within the line, when we are dealing with R2?
You can add any two vectors IN the subspace and stay in the subspace. Not any vector but any vector in the subspace. Try drawing a line that goes through the origin and pick two vectors on that line and add them. You will still be on that line because it is a subspace. I'm saying that if you add vectors u and v which are in the subspace W then u+v is a vector that is in W too. The reason that I am confined to the line is that the line is the subspace. The reason the line is a subspace is because I can add any two vectors in the subspace (i.e. on the line) and still be in the subspace. Also R2 is a vector space and a subspace (the line) is a part of the vector space that's been made into its own space because it follows the rules to be a space. That's why we call it a subspace.
Thank you for sharing my video. The minimum length (of the space) would be the simply the zero vector as all the vectors in the zero vector are zero and thus their sum is zero which is in the space. Though that's not much of a line. The only other possible size of the line is on that is infinitely long otherwise you could combine two vectors and not be in the space. Also if you had a non infinite line you could pick a vector u that is almost as long as the line and scale it by c and the vector cu would be out with the space. If you read the definition of a subspace carefully and think it through you should see what I mean.
Are there only three vector sub spaces within R2? or these are the basic three subspaces in R2? Can there not be subspaces defined apart from those three shown in this video?
+thetruereality R² and the zero vector are always subspaces of R². These are the two in the video. The third is a line through the zero vector, any line through the zero vector is a subspace. so you can turn it all you want, giving an infinite amount of subspaces in R². same for any dimension.
Before my question , lemme say this " Line is an instance of 2D. Many infinite amount of lines put together forms a plane which is 2D. Many infinite number of planes put together forms space which is 3D. So a plane is an instance of 3D. " My question is " 1) A subspace of R-n can only be entire R-n OR 2) An instance of R-n or an entire space of any subdimesional space of R-n = [R-(n-k)] OR 3) A Zero vector". 1,2,3 must pass through origin. So there can never be a subspace which is of order R-n which is a part of R-n like you showed in 2d, a part of entire plane of R2 ?
![What is a Subspace? [Passing Linear Algebra]](http://i.ytimg.com/vi/JoeVHP_jg44/mqdefault.jpg)
![What is a Subspace? [Passing Linear Algebra]](/img/tr.png)
This is amazing. Just walked out of a lecture at the University of Melbourne on subspaces, understood nothing. Within 2 minutes of your video, I learned more than a whole hour in the lecture. Thanks! Subscribed :D
Ouch that must say a lot about your school tbh xD
I usually dislike the way people try to explain certain concepts on TH-cam. This one was simple, clear to understand, and you summed it up nicely. Thank you for the post, and I agree that more videos from you would be excellent for those who are struggling on the subject matter!
This is the first time I saw a graphical representation of what a subspace is. I wonder why they don't show this in universities? Anyway thanks for the help, i understand it now.
Joseph Evora Interesting. Diagrams are always more helpful than text/equations. I'm glad this was helpful.
Joseph Evora I agree makes u wonder.
I know they give linear algebra on my university to filter out the bad students so I don't think they care if they can present something better.
Joseph Evora I noticed the same thing. I think it's stupid, because it really did help me understand the concept better. And of course I do realise that it isn't that easy for 4 or more dimensions but it's certainly helpfull when you're just starting with linear algebra.
I hear that a lot on the YT comments. What's worse is my prof wrote his own book and it doesn't even define a subspace, so here I am
Wdf? I have gone through many undergrad and a quite a few grad level courses which required the knowledge of what a subspace is, this is the first time it has ever made sense. Thank you!
kabascoolr hahahahah... Processing mode is ON
I LOVE this! I feel like everyone teaches linear algebra incorrectly, because everyone tries to teach definitions, never any graphs. But I feel like linear algebra should be taught more graphically like you just did, and it makes so much more sense this way. Thank you so much!
I really appreciate this. This video is a breath of fresh air after dealing with lectures that use too many abstractions that don't really get to the point at times.
Jesus, your vector space and subspace videos are by far the best. You've cleared everything up. Thank you
even a grade 6 student can understand this, I like how simple ,factual and short the explanation is. I stand and salute you
Mind blowing explanation .This is the only video which makes sense of subspace.
DAMMIT! A day before the finals and I am finding all these awesome videos....my professor makes everything so damn complicated for not apparent reason! Thanks!
Sleeping through lecture brought me to complete understanding of a subspace in no time at all. Thank you
Glad I can be of some help. It took me ages to understand it too until it all fell into place.
I was really getting headache learning about sub space from book..u explained it very easily and simply...thanks and keep uploading more such videos
The best explanation I've ever seen for subspace. Well done!
So much easier with the visual. I now get the zero vector rule with subspaces. Thank you.
agree :)
Thank you very much. I have my first midterm in linear algebra on Friday and the concept of subspaces was still a mystery to me. Your explanation was perfect and I finally understand it.
Thank you for the geometric representation. Please continue making these kinds of videos. It's really helpful
8 years later still great info. thanks!
It's nice how you showed examples of subsets of R^2 that are not subspaces of R^2.
I wish I saw this video sooner.. university makes this so confusing but it's a pretty simple concept. Thanks a lot!
Thank you! I was lost because I couldn't understand what a subspace was. Just in a few minutes I finally understood something that's been haunting me for a while. I couldn't go to class the day the topic was introduced in Algebra to make things worse.
my one month struggle and this man clears it in 7 mins ❤
I had the same problem. Really found subspaces hard until it suddenly made sense.
no wonder why this video has so many likes. Subscribed! :)
alhaleem Thank you.
Very simple and easy to understand. Helped me a lot in Linear Algebra.
thaaaaaaaank you. i hope that you would make more videos. you have a talent, you really do.
Best explanation of subspace so far
MANY MANY THANKS FOR WORKING HARD TO MAKE THIS TUTORIAL
Why do professors not explain this better lmao. Thank you, helped me understand a concept that a professor took an hour to not explain to me
Very good explanation with very good examples
Thanks soooo much man! i really wasn't getting the whole subspace thing until this vid. thanks sooo much!
This is so awesome!!! This video has just made my day!!! Now I know what I need to about subspaces!! Thank you so much for producing and posting it!! :) :) :) :)
I can not thank you enough, you are a freaking legend mate
this one is the best video that explain subspace
I love you whoever made this i owe you 1
Thank you for your kind comments. I found this very hard to learn at first
So helpful, I was picturing it wrong in my mind, thank you!
Thanks a lot for this video. Graphical representation helped a lot for me to understand the concept :)
So what are the subspaces of R3? I understand the 3 in R2 but at the end you said something about the subspaces in R3. Did you say a cube? Thanks
Very easy to understand your explanation!
Thank you so much for this video, was struggling to find the answer to a question I had and you made it very very clear!!!
thank goodness, now the subspace is much clearer to me now :)
because of you I got the entire idea, thank you so much
Very clear and concise, thank you for posting!!
Very informative😁 ...nice job. Lots of thanks and please put more videos on linear algebra.
Great work explaining this in a simple way!
Thank you! Very clear and concise. What's the application of subspaces, exactly? How are they useful?
Are we in the same math 270 class?
Probably!
You always sit in the front right? My name is Rip.
Sent from my iPhone
Yep. I sent you a message via Google+ hangout!
I'm completely new in linear algebra and I wonder what determines the size of the subspace?
o well sir you cleared my whole confusion for which i searched the whole youtube but cant overcome.....
Do subspaces need to go on to infinity then (whether a line, plane...)? So that adding two vectors will always land inside the subspace and not go outside of it.
great explanation, much love
Ugh, I can't believe I never understood this before... Thanks so much!
this was so nicely explained thanks a lot
For the second example, isn't that also not closed under addition? All the vectors should start at the origin and end up on the line, and then if you add two vectors from that line, you do not end up on the line. Am I misunderstanding?
I have question. Is it correct to oversimplify this: Can i say that a space(or plane?) is basically just bunch of dots with coordinates (depending on the dimension) and one vector is two selected random dots with distance and orientation between those two? So a subspace is random dots with specified range(either in a line of dots or something else covering bigger area of dots) and if one or both dots end out of that specified range it's not a subspace?
+TheiLame it's so easy to think this way but im not so sure if its correct :/ but it makes sense to these cases...
very helpful Andy Murray
so a zero vector is a threshold of a plan to enter or exit a different dimensions in plane vector?
Thanks for clearing the idea of subspace..
The best way to imagine subspace is as an upside down inverted cone normal space is the widest part of the cone while the smallest part is deepest part of subspace, Not unlike a black hole or a wormhole visually it's not because you can move through subspace, coordinates at the widest part of the cone are further apart then the coordinates at the the smaller side making travel between two points far shorter in subspace depending on how deep into subspace you can go.
You are not actually bending or compressing spacetime you are decompressing it around the object.
What do you mean (2 minutes) you talked about if there's no zero vector there, then there's a constant you can multiply by that will not be in the subspace? Not understanding what that means. Are you saying that if there's no zero vector then when you multiply by zero it will take you out of the subspace?
Laura W For any subspaces, zero vectors must be included regardless of whether the scalar is zero or not. If the zero vector is not in W then it is not a subspace.
saved my life ,,, thank u
thank you very much..i clearly understood...
This video really helped! Thanks a lot!
Good Job !
Very easy to understand (y)
Intuitively, a subspace would make sense to be defined as any space which is a subset of another space. But for some reason, mathematicians decided that a subspace also has to equal to the span of all vectors in the subspace.
And therefore, a subspace is basically an equal-to-or-lower dimentional version of the original space, centered at the origin.
In your 1st example of what "is" a subspace, you mention that you can add any two vectors and still be within the subspace of the line. How is this possible? and why must you only be confined to choosing vectors within the line, when we are dealing with R2?
You can add any two vectors IN the subspace and stay in the subspace. Not any vector but any vector in the subspace. Try drawing a line that goes through the origin and pick two vectors on that line and add them. You will still be on that line because it is a subspace.
I'm saying that if you add vectors u and v which are in the subspace W then u+v is a vector that is in W too.
The reason that I am confined to the line is that the line is the subspace. The reason the line is a subspace is because I can add any two vectors in the subspace (i.e. on the line) and still be in the subspace. Also R2 is a vector space and a subspace (the line) is a part of the vector space that's been made into its own space because it follows the rules to be a space. That's why we call it a subspace.
Thank you for your video. I have a question. For the line that goes through 0, what is the min or max size of the line?
It would have to be a line, which goes on forever, or you could get out of the space by multiplying a large scalar.
Thank you for sharing my video. The minimum length (of the space) would be the simply the zero vector as all the vectors in the zero vector are zero and thus their sum is zero which is in the space. Though that's not much of a line. The only other possible size of the line is on that is infinitely long otherwise you could combine two vectors and not be in the space. Also if you had a non infinite line you could pick a vector u that is almost as long as the line and scale it by c and the vector cu would be out with the space. If you read the definition of a subspace carefully and think it through you should see what I mean.
Came here searching for subspace in Star Trek. Ended up learning actual Math 😎
I second that
I third that. But still none the wiser
Came here searching for subspace emissary and this is my result .......
Are there only three vector sub spaces within R2? or these are the basic three subspaces in R2?
Can there not be subspaces defined apart from those three shown in this video?
+thetruereality R² and the zero vector are always subspaces of R². These are the two in the video. The third is a line through the zero vector, any line through the zero vector is a subspace. so you can turn it all you want, giving an infinite amount of subspaces in R². same for any dimension.
+The Benjamin thank you so much, very nice of you to reply
Really helpful. Cheers
thank you so much , this really helps to clear the concept!!!!
Thanks
Good teaching
Finally a decent simple explanation! Ta lad
best video ever
Well explained, thanks
Excellent!
Very helpful!
thank you, made it so simple
extremely helpful thanks!!
Nice, i understood, tanks a lot
Before my question , lemme say this " Line is an instance of 2D. Many infinite amount of lines put together forms a plane which is 2D. Many infinite number of planes put together forms space which is 3D. So a plane is an instance of 3D. "
My question is " 1) A subspace of R-n can only be entire R-n OR 2) An instance of R-n or an entire space of any subdimesional space of R-n = [R-(n-k)] OR 3) A Zero vector". 1,2,3 must pass through origin.
So there can never be a subspace which is of order R-n which is a part of R-n like you showed in 2d, a part of entire plane of R2 ?
My midsem exam is tomorrow. Thank you.
What is the difference between a subspace and a subset
omgggg I understand it ,god bless youuuuu
so helpful
superb
All of R2 means sir
my linear algebra teacher doesn't teach shit, thank you for this video
Love from India
thanks a lot.
thank you!
Thanks, my professor explained everything just like in the book and confused me lol.
Please suggest a reference book.
I like Introduction to Linear Algebra by Gilbert Strang
All we need now is an emissary
I still don't get it. Subspaces and Vector Spaces sound like the same thing!
Thank you for making this
thank you by brother thank yooooooou
That vector did not go through the origin .
Call me nitpicky but I clearly see it going above ;)
Am going to keep it simple, and starts out with equations 😅😅😅
im not gonna lie im here cuz the word was being said a lot in a fanfic and i had no idea what it meant
Thank you for making this video about vector spaces. My teacher is chinese and cant speak english at all. I HATE people who cant speak enlgish!!!!!!!
You should say I hate teachers who do not speak english b/c they should pass tests, but to say I hate people who dont speak english is very sad!
That's pretty fucked up.
Nice scottish accent! *