Linear combinations, span, and basis vectors | Essence of linear algebra, chapter 2

Linear combinations, span, and basis vectors | Essence of linear algebra, chapter 2


In the last video, along with the ideas of
vector addition and scalar multiplication, I described vector coordinates, where’s this back and forth between, for example,
pairs of numbers and two-dimensional vectors. Now, I imagine that vector coordinates were
already familiar to a lot of you, but there’s another kind of interesting way
to think about these coordinates, which is pretty central to linear algebra. When you have a pair of numbers that’s meant
to describe a vector, like [3, -2], I want you to think about each coordinate
as a scalar, meaning, think about how each one stretches
or squishes vectors, In the xy-coordinate system, there are two
very special vectors: the one pointing to the right with length
1, commonly called “i-hat”, or the unit vector in the x-direction, and the one pointing straight up, with length
1, commonly called “j-hat”, or the unit vector in the y-direction. Now, think of the x-coordinate of our vector
as a scalar that scales i-hat, stretching it by a factor of 3, and the y-coordinate as a scalar that scales
j-hat, flipping it and stretching it by a factor of 2. In this sense, the vectors that these coordinates
describe is the sum of two scaled vectors. That’s a surprisingly important concept, this
idea of adding together two scaled vectors. Those two vectors, i-hat and j-hat, have a
special name, by the way. Together, they’re called the basis of a coordinate
system What this means, basically, is that when you
think about coordinates as scalars, the basis vectors are what those scalars actually,
you know, scale. There’s also a more technical definition,
but I’ll get to that later. By framing our coordinate system in terms
of these two special basis vectors, it raises a pretty interesting, and subtle,
point: We could’ve chosen different basis vectors,
and gotten a completely reasonable, new coordinate system. For example, take some vector pointing up
and to the right, along with some other vector pointing down and to the
right, in some way. Take a moment to think about all the different
vectors that you can get by choosing two scalars, using each one to scale one of the vectors,
then adding together what you get. Which two-dimensional vectors can you reach
by altering the choices of scalars? The answer is that you can reach every possible
two-dimensional vector, and I think it’s a good puzzle to contemplate
why. A new pair of basis vectors like this still
gives us a valid way to go back and forth between pairs of numbers and two-dimensional vectors, but the association is definitely different
from the one that you get using the more standard basis of i-hat and
j-hat. This is something I’ll go into much more detail
on later, describing the exact relationship between different coordinate systems, but for right
now, I just want you to appreciate the fact that any time we describe vectors numerically,
it depends on an implicit choice of what basis vectors we’re using. So any time that you’re scaling two vectors
and adding them like this, it’s called a linear combination of those
two vectors. Where does this word “linear” come from? Why does this have anything to do with lines? Well, this isn’t the etymology, but one way
I like to think about it is that if you fix one of those scalars, and let the
other one change its value freely, the tip of the resulting vector draws a straight
line. Now, if you let both scalars range freely,
and consider every possible vector that you can get, there are two things that can happen: For most pairs of vectors, you’ll be able
to reach every possible point in the plane; every two-dimensional vector is within your
grasp. However, in the unlucky case where your two
original vectors happen to line up, the tip of the resulting vector is limited
to just this single line passing through the origin. Actually, technically there’s a third possibility
too: both your vectors could be zero, in which
case you’d just be stuck at the origin. Here’s some more terminology: The set of all possible vectors that you can
reach with a linear combination of a given pair of vectors is called the span of those two vectors. So, restating what we just saw in this lingo, the span of most pairs of 2-D vectors is all
vectors of 2-D space, but when they line up, their span is all vectors
whose tip sits on a certain line. Remember how I said that linear algebra revolves
around vector addition and scalar multiplication? Well, the span of two vectors is basically
a way of asking, “What are all the possible vectors you can
reach using only these two fundamental operations, vector addition and scalar multiplication?” This is a good time to talk about how people
commonly think about vectors as points. It gets really crowded to think about a whole
collection of vectors sitting on a line, and more crowded still to think about all
two-dimensional vectors all at once, filling up the plane. So when dealing with collections of vectors
like this, it’s common to represent each one with just
a point in space. The point at the tip of that vector, where,
as usual, I want you thinking about that vector with its tail on the origin. That way, if you want to think about every
possible vector whose tip sits on a certain line, just think about the line itself. Likewise, to think about all possible two-dimensional
vectors all at once, conceptualize each one as the point where
its tip sits. So, in effect, what you’ll be thinking about
is the infinite, flat sheet of two-dimensional space itself, leaving the arrows out of it. In general, if you’re thinking about a vector
on its own, think of it as an arrow, and if you’re dealing with a collection of
vectors, it’s convenient to think of them all as points. So, for our span example, the span of most
pairs of vectors ends up being the entire infinite sheet of two-dimensional
space, but if they line up, their span is just a
line. The idea of span gets a lot more interesting
if we start thinking about vectors in three-dimensional space. For example, if you take two vectors, in 3-D
space, that are not pointing in the same direction, what does it mean to take their span? Well, their span is the collection of all
possible linear combinations of those two vectors, meaning all possible vectors you get by scaling each
of the two of them in some way, and then adding them together. You can kind of imagine turning two different
knobs to change the two scalars defining the linear combination, adding the scaled vectors and following the
tip of the resulting vector. That tip will trace out some kind of flat
sheet, cutting through the origin of three-dimensional space. This flat sheet is the span of the two vectors, or more precisely, the set of all possible
vectors whose tips sit on that flat sheet is the span of your two vectors. Isn’t that a beautiful mental image? So what happens if we add a third vector and
consider the span of all three of those guys? A linear combination of three vectors is defined
pretty much the same way as it is for two; you’ll choose three different scalars, scale
each of those vectors, and then add them all together. And again, the span of these vectors is the
set of all possible linear combinations. Two different things could happen here: If your third vector happens to be sitting
on the span of the first two, then the span doesn’t change; you’re sort
of trapped on that same flat sheet. In other words, adding a scaled version of
that third vector to the linear combination doesn’t really give you access to any new
vectors. But if you just randomly choose a third vector,
it’s almost certainly not sitting on the span of those first two. Then, since it’s pointing in a separate direction, it unlocks access to every possible three-dimensional
vector. One way I like to think about this is that
as you scale that new third vector, it moves around that span sheet of the first
two, sweeping it through all of space. Another way to think about it is that you’re
making full use of the three, freely-changing scalars that you have at your disposal to access the full
three dimensions of space. Now, in the case where the third vector was
already sitting on the span of the first two, or the case where two vectors happen to line
up, we want some terminology to describe the fact
that at least one of these vectors is redundant—not adding anything to our
span. Whenever this happens, where you have multiple
vectors and you could remove one without reducing the span, the relevant terminology is to say that they
are “linearly dependent”. Another way of phrasing that would be to say
that one of the vectors can be expressed as a linear combination of the others since it’s
already in the span of the others. On the other hand, if each vector really does
add another dimension to the span, they’re said to be “linearly independent”. So with all of that terminology, and hopefully
with some good mental images to go with it, let me leave you with puzzle before we go. The technical definition of a basis of a space
is a set of linearly independent vectors that span that space. Now, given how I described a basis earlier, and given your current understanding of the
words “span” and “linearly independent”, think about why this definition would make
sense. In the next video, I’ll get into matrices
and transforming space. See you then!

100 comments

  1. I count myself as one of the luckiest people to have stumbled on this video just before taking linear algebra in college

  2. Understanding LA without visulisation is very difficult. Thank you for putting the efforts to make these videos. You are doing a great job.

  3. My dear! A lot of efforts behind this explanation & visuals. Thanks very much 🙏🙏🙏. As if vectors and scalars in a cartoon movie.

  4. Is he talking faster in these videos? I fell others have a different pace to them. Also slowing down the video makes it difficult to listen.

  5. 10 years ago when I was being taught all this in school I absolutely did not see the point of even knowing such information. Now I use it for work and thinking all the time.

  6. you are fucking amazing. I have to learn linear algebra for Machine Learning and I was sitting there DREADING it. But I'm actually THOROUGHLY enjoying it!!!! You need to make videos about your teaching philosophy! holy shit. I wish EVERY teacher were like you.

  7. Intuitively I felt that the span of 3 vectors should have described a 3d object. Am I correct? Or do I need to understand better?

  8. Hey I'm afraid your statement at the end of the video is incorrect. If I choose the L.D subset of R^2 given by A={(1,0),(0,1),(2,0)}, I cannot express (0,1) as a linear combination of (1,0) and (2,0). Your statement should be quantificated with 'there exists' instead of 'for all'. Great videos and best wishes

  9. Субботний вечер. Все друзья тусят по клубам, а я смотрю видео по линейной алгебре.

  10. @3Blue1Brown
    if all linear combinations for 2 vectors in a 3D space, give a span o a 2D plane projection as a result. Can we say then that a span for two vectors inside a 2D plane, is derived from the projection of an unknown 3D plane?

    Which is or might be the relationship between the 2 vectors and his parents?

    Maybe this is nonsense, just a thought. Give below your opinion 😀

  11. I study at one of germanys top engineering universities and your way of explaining this is so much more superior than my univeristy professors.

  12. great work. I never understood this stuff in school and turns out I lived fine without it, but it bothered me so much that I never did, that almost 30 years later decided to try again and found this. And now I do. 🙂 thanks.

  13. I literally left my Cornell math support tutoring crying, feeling worst, but some girl stopped me to direct me to your videos. Thank God for her and for your videos, bc I was on the verge of a breakdown <3

  14. U guys are rocking.. Pls come to tamilnadu.. In my school and in my college they didn't thought us to the real..

    Now I'm very interested a lot..Thanks to the channel, Youtube, Internet so many times ❤😍❤

  15. Let’s be honest: y’all watch this for the dialogue
    “I know this already”
    “Ah but young padawan, all knowing, you are not. A subtlety, there is.”

  16. Can someone please explain why aV + bW+ cU = 0 and U != aV + bW are equivalent (or point me to a comment that goes over this)?

  17. I wish these kinds of visuals were available when I was in high school. Math explained with animation in 3D space is sooo much easier to understand. I am going to share this with my kids and I hope they appreciate what you have done. Great Job!!!

  18. With this tool of coding, Grant, I sugest simplexes would be a nice future theme. From simplexes, it's possible to develop probability, topology, otimization, economics, etc. Congtatulations for your work.

  19. My knowledge before watching this video: Span of linearly dependent vectors ; My knowledge after watching this video: Span of linearly independent vectors

  20. Simply awesome graphical explanation !!! Can I know the name of the application/software you use for animation?? Please!!

  21. One 10 minute video explained the basics of Linear Algebra better than an entire semester at college. Thanks Grant!

  22. I did not understand why the last definition was relevant, seemed like a collection of words, but it seems it me its because im missing something. What am i missing?

  23. I do know who you are and I love you! I was introduced to you and the world of fun math by the best high school Mathletes teacher who recently moved schools. Thanks for your awesome videos!

  24. The way you depict the concepts you talk about is very on point! Great job! You make me want to learn more about Linear Algebra. As a Math undergraduate I really appreciate that!

  25. I see the correlation between how you described basis and the technical definition of basis. It’s that in the technical version, each linear independent vectors mentioned are the i-hat & j-hat you mentioned in your meaning of basis; the scalars spanning the full space in the linear combination they are composed of in the technical definition are the coefficients of each basis vectors, in your interpretation, that are stretching and shrinking the linear independent vectors in the linear combination.

  26. Dude I love your videos. Please please PLEASE do something like this for Stochastic Analysis. I am doing a masters degree in Mathematical Finance and I would love to see a beautiful explanation from building measure spaces to Itô's Formula.

  27. I have been finding basis and if a set of vectors is linearly independent or not but never really visualised what was going on. Just wow.

  28. In 2D we can call the span of a function that can exist on the x axis as the Domain, and the span of it that can exist on the y axis as the Image. This is how it works in French at least. Are there similar terms for a function's ability to exist on a space in 3D?

  29. It's odd to me that before I saw any video explaining what the symbols for vectors were I called the "hat" a hat. Just an example of how intuitive this system can be I guess.

  30. This is brilliant! I knew about these things from my studies but still watched all the way to the end because this video is qualitiy work and I enjoyed. Way to go! (Y)

  31. 8:48 would be useful to add clarification here that the rule does not apply if the "u" vector is a null vector, such as [0,0] because you can always multiply vectors with zero.

  32. 10 minutes and you managed to explain this better, simpler and more clear than my lecturer did in two 45 minute lectures- Thank you!!

  33. You gave me a second life.. I'm passionate about ML and going through every steps I found that my conception on linear algb,calculus isn't that strong to grasp the ideas of using different mathematical tools for ML. Now I'm able to get that and apply that. You are the savior

  34. Not really sure how I ended up here but this is amazing. This is my jam. Never studied this before. Is it normal to just close my eyes and visualize all this and it makes perfect sense?

  35. I wish I found such videos back when I was being stuffed with each concept the wrong way. My love for math would grow more and more and will finally become something that can benefit me in real life. I am 35 years old, do you guys think there still time to set back and understand them again this way so I could use them the way they meant to be used?

  36. Gosh! I know now why I didn't understand shit in those math classes in school. I NEED THESE VISUAL REPRESENTATIONS USING ANIMATIONS TO UNDERSTAND THE FUCKING MATH! And ofcourse, a tutor like this one and passion to learn.

  37. حضرت محاضره الشعاع في جامعتي بالمانيا و استغرقت المحاضره 3 ساعات و شرح ما بعد المحاضره 3 ساعات و عدت المحااضره والشرح مره ثانيه بالبيت و مافهمت شرحهن لان الكلام كان كتييييييير ومافي شرح واضح .الدكتور نفسه حافظ المحاضره و مافهماتها .لهيك بحب اشكركن على شرحكن يلي اكل من وقتي عشرين دقيقه مع الانتهاء بنتيجه

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