Main content

## Algebra (all content)

### Course: Algebra (all content) > Unit 20

Lesson 10: Matrices as transformations- Transforming vectors using matrices
- Use matrices to transform 3D and 4D vectors
- Transforming polygons using matrices
- Transform polygons using matrices
- Matrices as transformations
- Matrix from visual representation of transformation
- Visual representation of transformation from matrix
- Understand matrices as transformations of the plane

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Visual representation of transformation from matrix

Sal finds the drawing that appropriately represents the effect of a given 2x2 transformation matrix on the plane. Created by Sal Khan.

## Want to join the conversation?

- So, if I try to generalize the properties we observe here, can we say the following ?

- the upper left entree of the transformation matrix corresponds to a scaling on the x-axis

- the upper right entree of the transformation matrix corresponds to a translation on the x-axis

- the lower left entree of the transformation matrix corresponds to a translation on the y-axis

- the lower right entree of the transformation matrix corresponds to a scaling on the y-axis(103 votes)- Yes, the translation of the points along either the x or y axis is referred to as 'shearing.'

Look at pages 252 - 260 to see a table of these linear transformations. You can do a lot of things to the initial image using linear transformation matrices.

http://books.google.com/books?id=YmcQJoFyZ5gC&pg=PA251&lpg=PA251&dq=table+linear+matrix+transformation+operators&source=bl&ots=3wXIzMe9pq&sig=W3j0rlLlkZsHY2lYwPAyB5k9too&hl=en&sa=X&ei=h25ZU9GZEJWksQSs1YCIAQ&ved=0CFQQ6AEwBQ#v=onepage&q=table%20linear%20matrix%20transformation%20operators&f=false(30 votes)

- Could you also think of this as 3*(identity matrix)*(quadrilateral matrix)= 3*(quadrilateral matrix)?(33 votes)
- Technically Yes!

3(I) M = 3 IM = 3M

Therefore, the matrix M is being multiplied by a scalar 3, making the coordinates of the quadrilateral 3 times of its original number, expanding the area.(10 votes)

- This isn't exactly about the video, but how do you rotate a shape using a matrix? The scaling and reflection make intuitive sense, but I don't understand a general way to rotate a shape using a matrix.(13 votes)
- He needs to, as I have the same question. The video describes the process in one way without providing an intuitive means of reversing the scenario.(2 votes)

- How would it be possible to scale a quadrilateral whose points all lie in the positive quadrant?(5 votes)
- The transformation from this video will scale up a quadrilateral 3x in lengths, no matter what quadrants it lies in.

You might not like that though, because maybe you would prefer the center of the polygon to stay in place. If so, you would need to subtract a constant from all the points to shift it back to its original center.(6 votes)

- Are there any more in-depth videos on this topic? Sal describes a simple trial and error process in this video, which works fine by picking arbitrary points and testing. But, what about when you have to go the other direction to solve for the transformation matrix itself. This is a more common problem, and I don't see any material an how to do this conceptually. The problem "Hints" are pretty bad too...(7 votes)
- To solve matrix transformation, use this way:

1) write the coordinates of the original figure in a matrix like

x1 x2 x3 x4

y1 y2 y3 y4, if the coordinates are (x1,y1), (x2,y2) (x3,y3) and (x4,y4)

2) Mutiply the transformation matrix to the matrix written. So if the transformation matrix is

a b

c d

then you should mutiply like this

[a b][x1 x2 x3 x4]

c d y1 y2 y3 y4

3) The result is a 2-by-4 matrix, which contains the coordiates of the transformed figure(1 vote)

- Wouldn't it be easier if you realised that the matrix is really just 3(I), and if IA=A, then 3AI=3A?(4 votes)
- Yes, you 're right about the identity matrix.

3 0

0 3

is the same as 3 . I

But Sal doesn't do it that way as this was a special case whereas he needs to generalize it for everyone and every problem of this type.(3 votes)

- so, assume that I have a transformation matrix (rotation) T, and another regular matrix A.

multiplying TxA isn't the same as AxT, the former applies rotation, but what about the latter ?(3 votes)- Without any other information about "regular" matrix
`A`

, all you can say is that`A·T`

is the matrix multiplication of them.

If`A`

were also a transformation matrix, then`A·T`

would also be a transformation matrix that would apply both transformations, first the one from`T`

, then the one from`A`

.(4 votes)

- Since matrices aren't cumulative, is there a convention for the order in which position vectors should be placed when grouping them into a matrix?(2 votes)
- It doesn't matter. So long as you know which vector represents which point, you can put them in any order you want. If you look at the example and imagine swapping some columns, the result will be the same, but with the same columns swapped.(3 votes)

- I've tried to create a problem of my own and am not understanding the results I get. I created the following transformation matrix which should apply a 45 degree rotation:

| 1 -1 |

| 1 1 |

And Im trying to track where the vector (2, 1) lands. If I multiply that matrix transformation to this vector I get the vector (1, 3). That doesn't seem to be visually right. The original X vector (1, 0) became (1, 1) and right across it, diagonally was the vector Im tracking (2, 1). After the transformation I would assume it should have landed on (1, 2), not (1, 3). What am I doing wrong?(3 votes)- The Transformation matrix for 45 degree rotation is:

|cos 45 -sin 45|

|sin 45 cos 45|

The original horizontal unit vector i (1, 0) will lands on (cos 45, sin 45) when rotated 45 degree. Not (1, 1). Remember that the unit vector has a magnitude of 1. If it becomes (1, 1), it is wrong since that vector will have a magnitude of sqrt(2). Hope that helps!(1 vote)

- Does it matter in which order you multiply the transformation matrix with the position vector matrix?

Thank you(1 vote)- Since matrix multiplication is NOT commutative, Yes, the order matters.(3 votes)

## Video transcript

If the transformation matrix T is equal to three zero zero three, choose
which sketch can represent this transformation when applied
to the red quadrilateral. This is fascinating! They don't give us any coordinates,
save for the vertices of the quadrilateral, which are
really the most useful points to use when thinking about
potential transformations. So let's just make up some
just to see what would happen to the particular coordinates
that we're looking at. And I think that will give us enough information to think about this. I encourage you to do
it on your own first. Pause the video. Come up with some coordinates
for this red quadrilateral. Then, see what transformation
you get and which of these seem to be closest to
the one that you got. I'm assuming you've had a go at it. Let's just say for the sake of argument that this point right over here. That right there, you could
say that's the position vector. I'll represent it as a column vector. Let's say this is a square. So this is one comma one. I'll just write that as
a column vector one one. And let's say this one then
would be one negative one. One negative one. Then, this one over here
could be represented by the position vector
negative one negative one. Negative one negative one. Negative one negative one. Finally, this point
right over here could be represented by the position
vector negative one one. Negative one one. So let's see what the
transformation matrix would do when it transforms these four points. The way I'm gonna think about it, let me just take our
transformation matrix. So, three zero zero three. And I'm gonna multiply it
by a two by four matrix that represents all of
these position vectors. I'm gonna multiply it by,
we have this point one one. That's our first point. We have the point one negative one. One negative one. We have this point which
is negative one one. Negative one one. And then we have this point
which negative one negative one. Negative one negative one. These were convenient points to pick since they didn't give us the points because it'll make the math
fairly straightforward. What is this going to be equal to? We have a two by two. Two by two. Times a two by four. Matrix multiplication is defined here cause we have the same number of columns as we have rows, right over here. This is going to result
in a two by four matrix. This is going to give us
another two by four matrix, which makes sense, cause
we're going to need four column vectors here for our
four new transformed points. Let's figure out what it is. The first column vector, right over here. We can think about this
row and this column. Or actually, the first
position right over here the first row, first entry
is this row, this column. The second one's going to be the second row and the first column. So let's see, three times
one plus zero times one. Well that's three plus zero
so this is going to be three. Then over here, zero times
one plus three times one is going to give us three as well. I think you see already a pattern here. To get the the x-coordinate for each of these vectors, we're
involving this row. And this row, we really just multiplied the three times the x-coordinate here and then we don't involve the y-coordinate because we're multiplying it times zero. We'll see that over and over again. So over here you have three times one. Three times one plus
zero times negative one. So it's just really three
times one, which is three. Then over here you see
this for the y-coordinate each time we're only
involving the y-coordinate of the point before transformation. You see zero times one is zero. So we're essentially not
thinking about the x-coordinate. And then it's just three times negative one, which is negative three. You see what its doing, it's just scaling each of these up by a factor of three. Three times negative one is negative three plus zero times one,
so it's negative three. Then, it's going to be
zero times negative one is zero plus three times one is three and then finally, three
times negative one. Three times negative one plus zero times negative one is negative three. And zero times negative one plus three times negative one
is negative three again. So what's going to happen to it? Well, each of these
coordinates essentially get pushed out by a factor of three. Actually this one seems to be the closest to the one we're thinking about. How do I know that? Well look at this. This is the point one
comma one right over there. One comma one. It gets mapped to the
point three comma three. So, one two three. One two three. This is three comma
three right over there. This got mapped to that and
we see it with each of them. One negative one. One negative one gets mapped
to three negative three. Then, we have negative one one. Negative one one. Negative one one is getting
mapped to negative three three. Finally, of course,
negative one negative one. Negative one negative one gets mapped to negative three negative three. It's definitely the second. The second diagram is the one that represents the transformation matrix T being applied to the red quadrilateral.