Pop Video: Three-Dimensional Linear Transformations

Grant Sanderson • 3Blue1Brown • Boclips

Three-Dimensional Linear Transformations


Video Transcript

Hey folks! I’ve got a relatively quick video for you today, just sort of a footnote between chapters. In the last two videos, I talked about linear transformations and matrices, but I only showed the specific case of transformations that take two-dimensional vectors to other two-dimensional vectors.

In general throughout the series, we’ll work mainly in two dimensions, mostly because it’s easier to actually see on the screen and wrap your mind around. But more importantly than that, once you get all the core ideas in two dimensions, they carry over pretty seamlessly to higher dimensions. Nevertheless, it’s good to peak our heads outside of flatland now and then to, you know, see what it means to apply these ideas in more than just those two dimensions.

For example, consider a linear transformation with three-dimensional vectors as inputs and three-dimensional vectors as outputs. We can visualize this by smooshing around all the points in three-dimensional space, as represented by a grid, in such a way that keeps the grid lines parallel and evenly spaced and which fixes the origin in place. And just as with two dimensions, every point of space that we see moving around is really just a proxy for a vector who has its tip at that point, and what we’re really doing is thinking about input vectors “moving over” to their corresponding outputs. And just as with two dimensions, one of these transformations is completely described by where the basis vectors go. But now, there are three standard basis vectors that we typically use: the unit vector in the 𝑥-direction, 𝑖-hat; the unit vector in the 𝑦-direction, 𝑗-hat; and a new guy, the unit vector in the 𝑧-direction called 𝑘-hat.

In fact, I think it’s easier to think about these transformations by only following those basis vectors since the full 3D grid representing all points can get kinda messy. By leaving a copy of the original axes in the background, we can think about the coordinates of where each of these three basis vectors lands. Record the coordinates of these three vectors as the columns of a three-by-three matrix. This gives a matrix that completely describes the transformation using only nine numbers. As a simple example, consider the transformation that rotates space 90 degrees around the 𝑦-axis. So that would mean that it takes 𝑖-hat to the coordinates zero, zero, negative one on the 𝑧-axis. It doesn’t move 𝑗-hat so it stays at the coordinates zero, one, zero. And then 𝑘-hat moves over to the 𝑥-axis at one, zero, zero. Those three sets of coordinates become the columns of a matrix that describes that rotation transformation.

To see where vector with coordinates 𝑥𝑦𝑧 lands, the reasoning is almost identical to what it was for two dimensions: each of those coordinates can be thought of as instructions for how to scale each basis vector so that they add together to get your vector. And the important part, just like the 2D case, is that this scaling and adding process works both before and after the transformation. So to see where your vector lands, you multiply those coordinates by the corresponding columns of the matrix and then you add together the three results. Multiplying two matrices is also similar. Whenever you see two three-by-three matrices getting multiplied together, you should imagine first applying the transformation encoded by the right one then applying the transformation encoded by the left one. It turns out that 3D matrix multiplication is actually pretty important for fields like computer graphics and robotics since things like rotations in three dimensions can be pretty hard to describe, but they’re easier to wrap your mind around if you can break them down as the composition of separate, easier-to-think-about rotations.

Performing this matrix multiplication numerically is, once again, pretty similar to the two-dimensional case. In fact, a good way to test your understanding of the last video would be to try to reason through what specifically this matrix multiplication should look like, thinking closely about how it relates to the idea of applying two successive transformations in space.

In the next video, I’ll start getting into the determinant.

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