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## Precalculus

### Course: Precalculus > Unit 7

Lesson 7: Matrices as transformations of the plane- Matrices as transformations of the plane
- Working with matrices as transformations of the plane
- Intro to determinant notation and computation
- Interpreting determinants in terms of area
- Finding area of figure after transformation using determinant
- Understand matrices as transformations of the plane
- Proof: Matrix determinant gives area of image of unit square under mapping
- Matrices as transformations
- Matrix from visual representation of transformation

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# Proof: Matrix determinant gives area of image of unit square under mapping

Proof that the determinant of a 2X2 matrix gives the area of the image of the unit square under the mapping defined by the matrix. Created by Sal Khan.

## Want to join the conversation?

- what it means "unit square under mapping" ?(2 votes)
- A mapping is a function, or a transformation. They're all synonyms.

For a given point P in the domain, a function f(x) will send P to a particular point f(P) in the range. That point that it's mapped to is called the image of P, or the image of P under f.

Likewise for whole sets of points. If you have a set S of points in the domain, the set of points they're all mapped to is collectively called the image of S.

If you consider the set of points in a square of side length 1, the image of that set under a linear mapping will be a parallelogram. The title of the video says that if you find the matrix corresponding to that linear transformation, its determinant will be the area of that parallelogram.(6 votes)

- What does dialation mean?(1 vote)
- In common speech, 'dilate' means 'make bigger'. When the pupils of your eyes get larger, we say they are dilated.

In math, 'dilate' means to change the size of something (either bigger or smaller) in a uniform way. If I draw any two lines of the same length on my shape, then dilate the whole figure, the two lines will still be the same length as each other afterward. So if you stretch a square out into a rectangle, that is*not*a dilation, because it wasn't done uniformly.(3 votes)

- In the video at the2:46mark Sal says the ac area is ac/2. How does Sal get ac/2?(0 votes)

## Video transcript

- [Instructor] The goal of
this video is to feel good about the connection
that we've talked about between the absolute
value of the determinant of a two by two matrix and the area of the
parallelogram that's defined by the two column vectors of that matrix. So for example, I have this column vector right over here ac, so
that's this blue vector. So this distance right over here, it goes a in the x direction. So this distance right over here is a, and then it goes c in the y direction. So this distance right
over here is equal to c. And so this distance up
here is also equal to a, and this is also equal to c. So we have this vector and then we have the bd vector, and bd vector, so in the x direction it goes a distance of b right over there. Or if we draw it over here,
it goes a distance of b. And in the vertical direction, again it goes a distance of d. So this right over here is d and this distance right over here is d. And we can see that the
parallelogram created or defined by those two vectors, it's
area is right over there. Now let's see if we can
connect that to the determinant or the absolute value of the
determinant of this matrix. And we're just going to assume
for the sake of simplicity that a, b, c and d are positive values, although we can in the
future do this same thing where we had some other combination where some of them are not positive, but this will hopefully give you a clue of how we can prove it. Now, how can we figure out the
area of this parallelogram? Well, one technique would be find the area of this larger rectangle right over here, and then from that, subtract out the parts that
are not in the parallelogram. And so let's do that. So what's the area of
this larger rectangle? Let's see the dimensions here are, this length from here to here is a, and then from here to here is b. So this is a plus b on this side. And on this side up here, this part is d and then this
part is c right over here. So it's d plus c. So the area of the whole thing is going to be a plus b, times d plus c, which is equal to, we just do the distributive property a few times. It's going to be ad
plus ac plus bd plus bc. Now, from that we're going
to wanna subtract out all of these other parts that
are not in the parallelogram. So let's do that. So you have this triangle right over here whose area would be ac over
two, a times c over two, but you also have this one,
which has the same area. So if we subtract both of them out we'd wanna subtract out a total of ac. Each of those are ac over two. So to count both of them
let's subtract out an ac. And then of course we could
do these two triangles. And the area of each of these
triangles is bd over two, b times d over two, but add them together,
their combined area is bd. So let's subtract that out, minus bd. And now, what is the area
of this right over here? Well, that is b times
c, so minus b times c, actually and that's also the
area of this right over here. So we have another b times c, so minus 2bc. So let's see what's going on. So if we subtract these
out, that takes out that, that takes out that, and
if you take bc minus 2bc we're gonna be left
with just a negative bc. So all of this is going to be equal to ad, what we have there. Bc minus 2bc is just
gonna be a negativebc. Well, this is going to be the determinant of our matrix, a times d minus b times c. So this isn't a proof that
for any a, b, c, or d, the absolute value of
the determinant is equal to this area, but it
shows you the case where you have a positive determinant and all of these values are positive. So hopefully that feels
somewhat satisfying. And you can try, if you
like to prove the cases if you don't have a positive determinant or if some combination
of these are negative.