Let L denote the line through the origin in R2 with slope m. Show that reflection in L has matrix

Attachment 20415

Please help me do this exercise. Thanks so much.

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- Jan 12th 2011, 05:35 PMsshhExercise: Matrix transformations
Let L denote the line through the origin in R2 with slope m. Show that reflection in L has matrix

Attachment 20415

Please help me do this exercise. Thanks so much. - Jan 12th 2011, 05:47 PMAckbeet
What ideas have you had so far? Also, I assume you meant "reflection about L", right?

- Jan 12th 2011, 09:59 PMFernandoRevilla
A little help:

Por every find the symmetric point of with respect to . You'll obtain an expression of the form:

Fernando Revilla - Jan 13th 2011, 05:47 AMHallsofIvy
To find the matrix corresponding to a linear transformation from to , it is enough to see what it does to the "basis vectors", <1, 0> and <0, 1>.

Notice that

and than

so what the linear transformation does to <1, 0> and <0, 1> gives you the two columns of the matrix.

Now, the line, through the origin with slope m, y= mx, contains the points (0, 0) and (1, m) and so is in the direction of the vector <1, m>. To find the "reflection" of <1, 0> in that line, find its projection onto the vector <1, m>, then subtract that from <1, 0> to find its component perpendicular to the line. Subtract**twice**that vector from <1, 0> (subtracting once puts you**on**the line, subtracting again, on the other side) to find the reflection in the line. - Jan 13th 2011, 07:17 AMFernandoRevilla
:__Alternative 3__

Some courses of Linear Algebra cover the following and well known theorem:

If is a hyperplane of , then the projection matrix on with the usual inner product is:

As a consequence the simmetry matrix on is

In our case, so,

Fernando Revilla