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  • Matrix multiplication as composition | Essence of linear algebra, chapter 4

    70 بازدید

    Multiplying two matrices represents applying one transformation after another. Many facts about matrix multiplication become much clearer once you digest this fact. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you'r

  • Nonsquare matrices as transformations between dimensions | Essence of linear alg

    227 بازدید

    Because people asked, this is a video briefly showing the geometric interpretation of non-square matrices as linear transformations that go between dimensions. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you'r

  • The determinant | Essence of linear algebra, chapter 5

    69 بازدید

    The determinant of a linear transformation measures how much areas/volumes change during the transformation. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and

  • Inverse matrices, column space and null space | Essence of linear algebra, chapt

    25 بازدید

    How to think about linear systems of equations geometrically. The focus here is on gaining an intuition for the concepts of inverse matrices, column space, rank and null space, but the computation of those constructs is not discussed. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new v

  • Dot products and duality | Essence of linear algebra, chapter 7

    50 بازدید

    Dot products are a nice geometric tool for understanding projection. But now that we know about linear transformations, we can get a deeper feel for what's going on with the dot product, and the connection between its numerical computation and its geometric interpretation. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube,

  • Cross products | Essence of linear algebra, Chapter 8

    64 بازدید

    This covers the main geometric intuition behind the 2d and 3d cross products. *Note, in all the computations here, I list the coordinates of the vectors as columns of a matrix, but many textbooks put them in the rows of a matrix instead. It makes no difference for the result, since the determinant is unchanged after a transpose, but given how I've framed most of this series I think it is more intuitive to go with a column-centric approach. Watch the full "Essence of linear algebra" pla

  • Cross products in the light of linear transformations | Essence of linear algebr

    128 بازدید

    For anyone who wants to understand the cross product more deeply, this video shows how it relates to a certain linear transformation via duality. This perspective gives a very elegant explanation of why the traditional computation of a dot product corresponds to its geometric interpretation. *Note, in all the computations here, I list the coordinates of the vectors as columns of a matrix, but many textbooks put them in the rows of a matrix instead. It makes no difference for the result, since the deter

  • Change of basis | Essence of linear algebra, chapter 9

    30 بازدید

    How do you translate back and forth between coordinate systems that use different basis vectors? Watch the full "Essence of linear algebra" playlist here: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

  • Eigenvectors and eigenvalues | Essence of linear algebra, chapter 10

    99 بازدید

    A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more,

  • Abstract vector spaces | Essence of linear algebra, chapter 11

    124 بازدید

    The tools of linear algebra are extremely general, applying not just to the familiar vectors that we picture as arrows in space, but to all sorts of mathematical objects, like functions. This generality is captured with the notion of an abstract vector space. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want t

  • Essence of linear algebra preview

    106 بازدید

    This introduces the "Essence of linear algebra" series, aimed at animating the geometric intuitions underlying many of the topics taught in a standard linear algebra course. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive noti

  • Vectors, what even are they? | Essence of linear algebra, chapter 1

    36 بازدید

    I imagine many viewers are already familiar with vectors in some context, so this video is intended both as a quick review of vector terminology, as well as a chance to make sure we're all on the same page about how specifically to think about vectors in the context of linear algebra. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the dri

  • Linear combinations, span, and basis vectors | Essence of linear algebra, chapte

    78 بازدید

    The fundamental vector concepts of span, linear combinations, linear dependence, and bases all center on one surprisingly important operation: Scaling several vectors and adding them together. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to rec

  • Linear transformations and matrices | Essence of linear algebra, chapter 3

    111 بازدید

    Matrices can be thought of as transforming space, and understanding how this work is crucial for understanding many other ideas that follow in linear algebra. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're

  • Three-dimensional linear transformations | Essence of linear algebra, footnote

    100 بازدید

    What do 3d linear transformations look like? Having talked about the relationship between matrices and transformations in the last two videos, this one extends those same concepts to three dimensions. Watch the full "Essence of linear algebra" playlist here: https://goo.gl/R1kBdb ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the be