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

Lesson 10: Matrices as transformations

# 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
• 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.
• Could you also think of this as 3*(identity matrix)*(quadrilateral matrix)= 3*(quadrilateral matrix)?
• 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.
• 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.
• 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.
• How would it be possible to scale a quadrilateral whose points all lie in the positive quadrant?
• 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.
• 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...
• 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?
• 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.
• 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 ?
• 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`.
• 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?
• 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.
• 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?
• 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.