# Find the matrix of reflection through the line of equation ax + by = 0 in ℝ²

• August 31st 2013, 07:15 AM
twilightmage13
Find the matrix of reflection through the line of equation ax + by = 0 in ℝ²
Note: I'm using ";" to show in my matrices and vectors to represent the end of a row, e.g. [2; 4] is a column vector with 2 at the top and 4 at the bottom.

I know how to find the matrix of reflection through y = x in ℝ² (for which a = -b) by using the standard bases e1 = [1; 0], e2 = [0; 1] and got the matrix T = = [0 1; 1 0]
I tried exploring with y = -x (for which a = b) and got T = [0 -1; -1 0].

But I'm not seeing any pattern to a reflection across such a general case ax + by = 0.
• August 31st 2013, 09:10 AM
HallsofIvy
Re: Find the matrix of reflection through the line of equation ax + by = 0 in ℝ²
"Reflection in a line" maps any vector parallel to the line into itself. It maps any vector perpendicular to the line into its negative.

So: What is a vector in the direction of the line ax+ by? What is a vector in the direction of ax+ by= 0? What is a vector perpendicular to it?

Another way: find the rotation matrix, A, that rotates ax+ by= 0 to y= 0. The matrix that refleclects about y= 0 is $R= \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$. The matrix that reflects in the line ax+ by= 0 is $A^{-1}RA$.
• August 31st 2013, 07:23 PM
johng
Re: Find the matrix of reflection through the line of equation ax + by = 0 in ℝ²
Hi,
Please do not post a question more than once. The attachments provide a complete answer to your question as suggested by the previous response:

Attachment 29098
Attachment 29099