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[classical music]
"Lisa: Well, where's my dad?
Frink: Well, it should be obvious to even the most dimwitted individual who holds an advanced degree in hyperbolic topology that Homer Simpson has stumbled into
... (dramatic pause) ...
the third dimension."
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 for 3-D grid representing all
points can get kind of 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 3×3 matrix.

This gives a matrix that completely describes the transformation using only nine numbers.
As a simple example, consider,
the transformation that rotate space

90 degrees around the y-axis.
So that would mean that it takes i-hat
to the coordinates [0,0,-1]
on the z-axis,

it doesn't move j-hat so it stays at the
coordinates [0,1,0]

and then k-hat moves over to the x-axis at
[1,0,0].

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 2-D 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 3×3 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 3-D 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 of
transformations in space.

In the next video I'll start getting
into the determinant.

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Three-dimensional linear transformations | Essence of linear algebra, chapter 5 中文 (Three-dimensional linear transformations | Essence of linear algebra, chapter 5)

111 分類 收藏
Chun Sang Suen 發佈於 2018 年 8 月 23 日
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