# Animated mathSeptember 11, 2016 12:21 AM   Subscribe

Essence of linear algebra - "[Grant Sanderson of 3Blue1Brown (now at Khan Academy) animates] the geometric intuitions underlying linear algebra, making the many matrix and vector operations feel less arbitrary."

1. Vectors, what even are they? - "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." (9:48)
2. Linear combinations, span, and basis vectors - "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." (9:55)
3. Linear transformations and matrices - "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." (10:54)
4. Matrix multiplication as composition - "Multiplying two matrices represents applying one transformation after another. Many facts about matrix multiplication become much clearer once you digest this fact." (9:59)
fn. Three-dimensional linear transformations - "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." (4:42)
5. The determinant - "The determinant of a linear transformation measures how much areas/volumes change during the transformation." (9:59)
6. Inverse matrices, column space and null space - "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." (12:04)
fn. Nonsquare matrices as transformations between dimensions - "Because people asked, this is a video briefly showing the geometric interpretation of non-square matrices as linear transformations that go between dimensions." (4:22)
7. Dot products and duality - "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." (14:07)
8. Cross products - "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." (8:54)
part 2. Cross products in the light of linear transformations - "This covers the main geometric intuition behind the 2d and 3d cross products." (13:11)
also btw...
Eigenvectors and Eigenvalues explained visually - "Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's PageRank algorithm. Let's see if visualization can make these ideas more intuitive." (via)
posted by kliuless (17 comments total) 141 users marked this as a favorite

These are really nice. If somebody had told me when I started learning determinants that they're basically a way of representing how much transformations scale things I would have been a lot happier doing all that rote adding and subtracting. Similarly when I learned the formula for eigenvectors in undergrad and it wasn't until I did some Principal Component Analysis later on in a stats course that anybody bothered to explain to me what they might actually mean ever. Actually it looks like eigenvectors are the next video coming up so I'm looking forward to seeing his pretty animated take on it.
posted by eykal at 7:25 AM on September 11, 2016 [8 favorites]

Good list, I've looked at some of these before, good thing I've got a small screen as some of the spinning coordinate lines could cause sea sickness. Visualizing the meaning of matrix multiplication is much clearer in these than any other place but still twists my brain around.
posted by sammyo at 9:05 AM on September 11, 2016

The animations are nice but all this can be taught by a good teacher with a chalkboard, and it's the way that good teachers have probably taught it for a hundred years or more.

I know it's still done this way this century, because I've seen it taught that way then later done it myself. Colored chalk helps.
posted by SaltySalticid at 9:25 AM on September 11, 2016 [1 favorite]

Holy shit. As an engineering student 20 years ago, I took LA classes and even tutored other students well enough to get them through classes...without understanding, completely via rote memorization. Thank you for this!!!!
posted by notsnot at 9:41 AM on September 11, 2016 [5 favorites]

but all this can be taught by a good teacher with a chalkboard

Good teachers are a lot harder to find (and a lot more expensive) than watching YouTube animations for self-education, though.
posted by unknownmosquito at 11:31 AM on September 11, 2016 [8 favorites]

In undergrad physics I had brief reviews of linear algebra atleastt 4 times. I was really happy I'd seen the "Let's see how the matrix maps a unit cube into a space, ... and then the determinant is obviously just the volume described by these remapped unit vectors " framing back in high school math class.
posted by sebastienbailard at 1:48 PM on September 11, 2016 [1 favorite]

My high school class briefly touched on the brute computational 'how-to calculate' parts, but I had no idea what it was even useful for. I have occasionally thought of it and wondered, WTF was that even for? I've even gone so far as to google around and try to self-teach it, but still all numbers, numbers, shrug.

These videos are great.
posted by ctmf at 1:50 PM on September 11, 2016 [2 favorites]

In my experience, most mathematical instruction, except for the highest levels, seems to be of the 'how-to calculate' kind.
Usually, the 'why' part is not deemed necessary. (Now, I got my instruction before calculators were affordable, so that might have changed)

I had the good fortune of being part of the 'new math' instruction in the 60's, which tried to teach some of the theory, not just the memorization of times tables and steps. It was also fun because most of the teachers had no idea what was going on. I ended up trying to figure out the 'why' a lot.

I love this kind of visual explanation. I remember finding a calculus paperback in a free book pile that graphically showed me the 'why' of differentiation, 15 years after my first calc class, where I was only shown the 'how-to'.
posted by MtDewd at 2:42 PM on September 11, 2016 [2 favorites]

this can be taught by a good teacher with a chalkboard

The thing is that virtually everybody has a less than optimal teacher. Education is way too important to leave it up to some random person your school assigned to teach (or "teach") your class. Everybody should be learning from the absolute best teacher for every subject and that means recorded media with individual support for localization and questions.
posted by LastOfHisKind at 2:49 PM on September 11, 2016 [6 favorites]

And even the best teacher is not necessarily a perfect match for every (good) student, having different approaches available is another great tool.
posted by sammyo at 3:36 PM on September 11, 2016 [3 favorites]

Oh yes this is awesome! I was just talking to a friend the other day how obtuse I found linear algebra. I'm greatly looking forward to looking through these, because I apparently had a horrible linear algebra teacher back in college, seriously.

I'm glad my undergrad Calculus teacher, good ol' Dr. Balman, spent a lot of time on the why-fors of Calculus, which I understood pretty well, even if my efforts to apply them on tests inevitably make a mistake one place or another. It is odd... I feel now, in my 40s and with a Masters in English Lit, that I can probably do the math a lot better than when I was an undergrad!
posted by JHarris at 3:44 PM on September 11, 2016 [1 favorite]

Looking forward to digging into these, thank you for the post!
posted by invokeuse at 8:03 PM on September 11, 2016

What I've watched so far (determinants) are an excellent display of intuitions — the beauty of math.
posted by esprit de l'escalier at 8:49 PM on September 11, 2016 [1 favorite]

Math 212, linear algebra, was the only math course I took in college (first semester freshman year) and turned me off entirely to a subject I really enjoyed. Should have taken DiffEq or something else. Because there was no why, no meaning. The eigenvalue link still doesn't help.

Now, it hurt me in terms of handling co-ordinate systems and rotations/projections, but ..
posted by k5.user at 8:27 AM on September 12, 2016

It sounds like I am not alone in having had a pretty bad college Linear Algebra class. Mine was basically, "Here are some theorems you can prove about vector spaces," plus some of the important algorithms for numerical computation.

Completely missing was any hint of:

1. This shit is incredibly important in about a million applied fields, seriously you will probably use this a million times if you do applied statistics. (This was before stats / ML got hot, but still.)
2. Geometric intuitions for transformations, eigenvectors / eigenvalues, etc.
posted by grobstein at 9:43 AM on September 12, 2016 [1 favorite]

I had several excellent, passionate math teachers, but the first 90 seconds of that video was like someone putting a wad of paper under the wobbling bar stool you've sat on all evening. I understand how it's difficult to include, but I starved for the why. I mean the first complaint of any school kid about a topic is that they'll never need the skills. Without building a case for the need it's like teaching people the exact movements to chisel every individual beam of a house but never telling them what the hell all these blisters are for.
posted by lucidium at 5:45 PM on September 12, 2016 [2 favorites]

Following the eigenvectors link led me to the MIT Open Courseware 18.06 Linear Algebra video lecture series. It's quite good (and accessible for "some math familiarity" people like myself). Old-skool chalkboard.
posted by ctmf at 8:21 PM on September 18, 2016 [1 favorite]

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