my problem came about when applying a 2x2 matrix to a hyperbola..I'm fine with transforming point what I have trouble with is transforming lines.Theres two types I'd like help with:(A) : Given a line set to a constant...like y = a, how to I apply this to a matrix? I was thinking of making a column vector with the top part set to zero and the bottom set to a and applying this to the transformation matrix.(B) : For that hyperbola example, it was a simple unit hyperbola with its asymptotes : y=x, y = -x.How do I intepret what these transformations do to lines, with respect to arbitrary constants like y=a and for variables.

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my problem came about when applying a 2x2 matrix to a hyperbola..I&#039;m fine with transforming point what I have trouble with is transforming lines.Theres two types I&#039;d like help with:(A) : Given a line set to a constant...like y = a, how to I apply this to a matrix? I was thinking of making a column vector with the top part set to zero and the bottom set to a and applying this to the transformation matrix.(B) : For that hyperbola example, it was a simple unit hyperbola with its asymptotes : y=x, y = -x.How do I intepret what these transformations do to lines, with respect to arbitrary constants like y=a and for variables.

soymmernenx · Accepted Answer

linear transformations map lines onto lines or collapse them onto the origin. However, you must take care that you have the right representation of the line you’re transforming. A single vector suffices to represent a line through the origin, but in general you’ll need two-one (d) to give the direction of the line and another that gives a point p on the line. They combine to form a parametric representation of the line: p+td. Both of these vectors need to be transformed to find the line’s image. It should be obvious why for a line through the origin the direction vector is enough since you can always choose p=0 for it.For example, the vector   (  0  ,  a      )    T   doesn’t represent the line   y  =  a. You instead need something like   (  0  ,  a      )    T    +  t  (  1  ,  0      )    T    =  (  t  ,  a      )    T  . The image of this line under the transformation   M will be   M  (  0  ,  a      )    T    +  t  M  (  1  ,  0      )    T    =  M  (  t  ,  a      )    T  For part (B), both of those lines go through the origin, so it’s enough to transform a representative vector for each of these lines to find their images.

my problem came about when applying a 2x2 matrix to a hyperbola..I'm fine with transforming point wh

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