Hello, hello again.
So, moving forward
I will be assuming you have a visual understanding of linear transformations
and how they're represented with matrices
the way I have been talking about in the last few videos.
If you think about a couple of these linear transformations
you might notice how some of them seem to stretch space out
while others squish it on in.
One thing that turns out to be pretty useful to understanding one of these transformations
is to measure exactly how much it stretches or squishes things.
More specifically
to measure the factor by which the given region increases or decreases.
For example
look at the matrix with the columns 3, 0 and 0, 2
It scales i-hat by a factor of 3
and scales j-hat by a factor of 2
Now, if we focus our attention on the one by one square
whose bottom sits on i-hat and whose left side sits on j-hat.
After the transformation, this turns into a 2 by 3 rectangle.
Since this region started out with area 1, and ended up with area 6
we can say the linear transformation has scaled it's area by a factor of 6.
Compare that to a shear
whose matrix has columns 1, 0 and 1, 1.
Meaning, i-hat stays in place and j-hat moves over to 1, 1.
That same unit square determined by i-hat and j-hat
gets slanted and turned into a parallelogram.
But, the area of that parallelogram is still 1
since it's base and height each continue to each have length 1.
So, even though this transformation smushes things about
it seems to leave areas unchanged.
At least, in the case of that one unit square.
Actually though
if you know how much the area of that one single unit square changes
it can tell you how any possible region in space changes.
For starters
notice that whatever happens to one square in the grid
has to happen in any other square in the grid
no matter the size.
This follows from the fact that grid lines remain parallel and evenly spaced.
Then, any shape that is not a grid square
can be approximated by grid squares really well.
With arbitrarily good approximations if you use small enough grid squares.
So, since the areas of all those tiny grid squares are being scaled by some single amount
the area of the blob as a whole
will also be scaled also by that same single amount.
This very special scaling factor
the factor by which a linear transformation changes any area
is called the determinant of that transformation.
I'll show how to compute the determinate of a transformation using it's matrix later on
in the video
but understanding what it is, trust me, much more important than understanding the computation.
For example the determinant of a transformation would be 3
if that transformation increases the area of the region by a factor of 3.
The determinant of a transformation would be 1/2
if it squishes down all areas by a factor of 1/2.
And, the determinant of a 2-D transformation is 0
if it squishes all of space onto a line.
Or, even onto a single point.
Since then, the area of any region would become 0.
That last example proved to be pretty important
it means checking if the determinant of a given matrix is 0
will give away if computing weather or not the transformation associated with that matrix
squishes everything into a smaller dimension.
You will see in the next few videos
why this is even a useful thing to think about.
But for now, I just want to lay down all of the visual intuition
which, in and of itself, is a beautiful thing to think about.
Ok, I need to confess that what I've said so far is not quite right.
The full concept of the determinant allows for negative values.
But, what would scaling an area by a negative amount even mean?
This has to do with the idea of orientation.
For example
notice how this transformation
gives the sensation of flipping space over.
If you were thinking of 2-D space as a sheet of paper
a transformation like that one seems to turn over that sheet onto the other side.
Any transformations that do this are said to "invert the orientation of space."
Another way to think about it is in terms of i-hat and j-hat.
Notice that in their starting positions, j-hat is to the left of i-hat.
If, after a transformation, j-hat is now on the right of i-hat
the orientation of space has been inverted.
Whenever this happens
whenever the orientation of space is inverted
the determinant will be negative.
The absolute value of the determinant though
still tells you the factor by which areas have been scaled.
For example
the matrix with columns 1, 1 and 2, -1
encodes a transformation that has determinant
Ill just tell you
-3.
And what this means is
that, space gets flipped over
and areas are scaled by a factor of 3.
So why would this idea of a negative area scaling factor
be a natural way to describe orientation flipping?
Think about the seres of transformations you get
by slowly letting i-hat get closer and closer to j-hat.
As i-hat gets closer
all the areas in space are getting squished more and more
meaning the determinant approaches 0.
once i-hat lines up perfectly with j-hat,
the determinant is 0.
Then, if i-hat continues the way it was going
doesn't it kinda feel natural for the determinant to keep decreasing into the negative numbers?
So, that is the understanding of determinants in 2 dimensions
what do you think it should mean for 3 dimensions?
It [determinant of 3x3 matrix] also tells you how much a transformation scales things
but this time
it tells you how much volumes get scaled.
Just as in 2 dimensions
where this is easiest to think about by focusing on one particular square with an area 1
and watching only what happens to it
in 3 dimensions
it helps to focus your attention
on the specific 1 by 1 by 1 cube
whose edges are resting on the basis vectors
i-hat, j-hat, and k-hat.
After the transformation
that cube might get warped into some kind of slanty slanty cube
this shape by the way has the best name ever
parallelepiped.
A name made even more delightful when your professor has a nice thick Russian accent.
Since this cube starts out with a volume of 1
and the determinant gives the factor by which any volume is scaled
you can think of the determinant
as simply being the volume of that parallelepiped
that the cube turns into.
A determinate of 0
would mean that, all of space is squished onto something with 0 volume
meaning ether a flat plane, a line, or in the most extreme case
onto a single point.
Those of you who watched chapter 2
will recognize this as meaning
that the columns of the matrix are linearly dependent.
Can you see why?
What about negative determinants?
What should that mean for 3 dimensions?
One way to describe orientation in 3-D
is with the right hand rule.
Point the forefinger of your right hand
in the direction of i-hat
stick out your middle finger in the direction of j-hat
and notice how when you point your thumb up
it is in the direction of k-hat.
If you can still do that after the transformation
orientation has not changed
and the determinant is positive.
Otherwise
if after the transformation it only makes since to do that with your left hand
orientation has been flipped
and the determinant is negative.
So if you haven't seen it before
you are probably wondering by now
"How do you actually compute the determinant?"
For a 2 by 2 matrix with entries a, b, c, d
the formula is (a * d) - (b * c).
Here's part of an intuition for where this formula comes from
lets say the terms b and c both happed to be 0.
Then the term a tells you how much i-hat is stretched in the x direction
and the term d
tells you how much j-hat is stretched in the y direction.
So, since those other terms are 0
it should make sense that a * d
gives the area of the rectangle that our favorite unit square turns into.
Kinda like the 3, 0, 0, 2 example from earlier.
even if only one of b or c are 0
you'll have a parallelogram
with a base a
and a height d.
So, the area should still be
a times d.
Loosely speaking
if both b and c are non-0
then that b * c term
tells you how much this parallelogram
is stretched or squished in the diagonal direction.
For those of you hungry for a more precipice description of this b * c term
here's a helpful diagram if you would like to pause and ponder.
Now if you feel like computing determinants by hand
is something that you need to know
the only way to get it down is to just
practice it with a few.
There's not really that much I can say or animate that is going to drill in the computation.
This is all tripply true for 3-rd dimensional determinants.
There is a formula [for that]
and if you feel like that is something you need to know
you should practice with a few matrices
or you know, go watch Sal Kahn work through a few.
Honestly though
I don't think those computations fall within the essence of linear algebra
but I definitely think that knowing what the determinate represents
falls within that essence.
Here's kind of a fun question to think about before the next video
if you multiply 2 matrices together
the determinant of the resulting matrix
is the same as the product of the determinants of the original two matrices
if you tried to justify this with numbers
it would take a really long time
but see if you can explain why this makes sense in just one sentence.
Next up
I'll be relating the idea of linear transformations covered so far
to one of the areas where linear algebra is most useful
linear systems of equations
see ya then!