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

The fundamental concepts of span, linear combinations, linear dependence, and bases.
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Arabic: @Cewkins, Hazem
Hebrew: Omer Tuchfeld
Spanish: Juan Carlos Largo
Vietnamese: @ngvutuan2811

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[lbblackburn]

I don't know who you are, but these linear algebra videos are brilliant. They are pedagogically invaluable and should be incorporated into every introductory linear algebra course. I teach linear algebra and I mention these visualizations but my hand-drawn figures on the marker board, my clumsy gestures in the air, and the textbook's static graphics are all quite inadequate for most students. I will be directing my students to these videos in the future or even playing them in class. Thank you.

[astronemir]

I'm an astrophysics phd and I use linear algebra everyday but I'm here watching these videos because they are so intuitive...

[erenyeagar]

I am still in 12th grade and this is all I had on Google Play rewards money. But as soon as I come of age(later this year), I will save up money to send you dude. Stay at it bro

[mechabunnyc]

1. a coordinate is a scalar, which scales the basis vector of its coordinate system
2. anytime we describe a vector numerically, it depends on the basis vectors we are using
3. a linear combination of a set of vectors is to scale them and add them together
4. the span of a set of vectors is the set of all linear combinations of the vectors
5. if a vector is a linear combination of a set of vectors, the vectors are linearly dependent
6. if each vector adds another dimension to the span, the vectors are linearly independent
7. the span of two linearly independent vectors is the 2D space, if the two vectors line up, their span is a certain line
8. when thinking about one vector, think of it as an arrow, when thinking about a collection of vectors, think of them as points
9. in three-dimensional space, the span of two linearly independent vectors is an infinite flat sheet, the span of three linearly independent vectors is the 3D space, if linearly dependent, the span is still a flat sheet
10. the basis of a vector space is a set of linearly independent vectors that span the full space

[KCHuang]

So many years of 'rigorous' linear algebra, but I still didn't have a good understand of the intuition behind it. Grant, you are a miracle worker. So happy to see you ended up in the math education field! fsc <3

[adamsubora]

Found this series 5 days after taking my Linear Algebra final. It's nice to finally understand what was going on the whole semester lol

[Karrismx]

Thank you! This is probably one of the most beautifully explained videos ever, your voice and animations are incredibly helpful to understand and enjoy the video👏🏼💐

[sebastianmonsalvo]

MIT undergrad here. Your video just taught me in 9 minutes what my math professor and teaching assistants couldn't in the past 2 weeks. You're amazing thank you!!

[anderslauridsen]

The definition makes sense since a linearly independent vector (no matter how matter how many dependent vectors there are) unlocks exactly one new dimension. 1 vector can describe all of 1d space with a scalar of some kind, but only 1 space. Adding a linearly independent vector of that one unlocks another dimension and so on. If we were to add a linearly dependent vector we would not get a new dimension no matter how we scale it. awesome video btw (i hope my comment was readable)

[heartbrokendra]

I literally left my Cornell math support tutoring crying, feeling worse, 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

[santiagogonzalezirigoyen]

"as you scale that new third vector, the sheet moves through the entire 3D space”, that’s the kind of thinking that pushes people to understand the concept for themselves. Thank you so much for your videos!

[el-p]

I'm a school dropout and I could never understand math. I thought I was stupid and had no talent for it, until I found this channel, which to my surprise helps me understand! This is better than any math book I've ever tried to conquer. The video format bypasses my mental block which interferes when I sit down with pen and paper. I feel like these videos are teaching me the general process of math and its nature of problem solving, so with this I can finally learn to self-learn.

Education has made some amazing advancements, and (at least my local)school system seems to be lagging behind. I have never learned from teachers and homework, and writing with pen and paper. There must be more people like myself out there that need to be shown that there are alternate methods to learning which might suit them better.

[TuningFreak]

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

[badlydrawnturtle]

Complex Algebra: "We call the vertical unit i."
Linear Algebra: "We call the horizontal unit i."

[danielayoutube]

I would like to leave an appreciation for the fact that you start with something most students are familiar with, develop our intuition and finally provide the definition. School teachers please learn how it's done. I cannot stress enough how helpful your videos are, thank you! Greetings from Portugal

[jacobcarignan]

Holy crap. I have been hammering through my textbook and lecture notes on spans, linear dependence/independence, and basis, and I feel like I’ve had my mind blown by the intuition you gave me by showing the math graphically. Everything makes so much more sense. Normally I don’t comment like this on other tutoring videos but this is soooo helpful

[lohnthom]

The peaceful music really helps set aside the onset of anxiety that usually comes at sight of numbers and equations.

[madhusai]

I don't know who you are, but I will find you and I will thank you

[Zephyr-tghu]

I'm just gonna binge this tonight instead of netflix haha

[avinashmaurya]

The basis of a vector space is a set of linearly independent vectors whose linear combinations can span the whole vector space.