Transforming a line segment with matrices?

[mneox]

Homework Statement

If you have the matrix X = [ 1 1; 1 0] and Y = [ 0 -1; 1 0] and line segment A = {(0,y) | 0<y<1}

Draw the images of line segment after you transform it by matrix X, Y, XY, YX
(the image of A after transformation by linear transformation of L is {L(a) | all a in A}

Homework Equations

T = A(x)
not sure if this is applicable here...

The Attempt at a Solution

what are you supposed to do for this? I'm confused.. thanks!

what are "images"? and how do you transform with matrices.. is it just multiplying?

sorry i am just confused really badly :s
thanks for your help and time

 

[Filip Larsen]

Yes, transformation of a point by matrix means you have to multiply the matrix with the radius-vector for the points in the line segment. When drawing the images, it may help you to know that since the transformation is linear the line segment will always be transformed into another line segment, so you only really have to transform the end points to get the image.

 

[mneox]

Yes, transformation of a point by matrix means you have to multiply the matrix with the radius-vector for the points in the line segment. When drawing the images, it may help you to know that since the transformation is linear the line segment will always be transformed into another line segment, so you only really have to transform the end points to get the image.

Thanks for the reply!

I see, but which part is this radius-vector though? Is it the (0,y) from A = {(0,y) | 0<y<1}? How do you know what value of y to use..?

Could you explain what A = {(0,y) | 0<y<1} even means?

I feel like I'm on the verge of grasping this but at the same time I'm lost. Thanks.

 

[Filip Larsen]

What points do (0,y) "translate" to when y varies from 0 to 1? Can you plot those points in a diagram? You should get a line segment, that is, a line with a start and end point. Can you give the coordinates of those end-points?

Notice, that with the y interval defined as it is with 0<y<1, the end-points themselves are not considered part of the open line segment, that is, the set A is the line segment between the two end-points but excluding the end-points themselves. You can treat them like normal end-points if you just remember that the are excluded from the images as well.

 

[HallsofIvy]

The "radius vector" Filip Larson mentioned in his first post is probably the "position vector" of a point which is the vector from (0, 0) to the point. It's components are the same as the coordinates of the point. That is, the position vector for (0, 0), at one end of the segment, is just the 0 vector= <0, 0> and the position vector of (0, 1), at the other end is $\vec{i}$= <0, 1>.

The line segment X maps A into is the line segment having endpoints
$$\begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix}0 \\ 0\end{bmatrix}$$
and
$$\begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix}0 \\ 1\end{bmatrix}$$