Question

How do we find a standard Matrix of a reflection in R^2 in the line 2x + 7y = 0?

Answer 1

As ever, find the image of the two unit vectors: i and j. Once you know those, those will be the columns of the matrix.

Answer 2

Sorry I'm not sure what you mean. We saw a method where we have refl_n(x) (where n = normal, and x = vector) = x - 2(proj_n(x))

Answer 3

I thought we'd have to try and find the normal but I'm not sure how :S?

Answer 4

Right, that's the way you find the image of any one vector. But let's get this first principle clear. Let T be a transformation matrix on R^2, say
a b
c d

Then notice when you multiply that matrix with the column vector (1 0)^t, you get the column vector (a c)^t. In other words, the first column of the matrix is the image of the unit vector i under the transformation

Answer 5

Likewise the column vector (b d)^t is the image of the unit vector j.

Answer 6

Hence for any linear transformation on R^2, you can write down the matrix for the transformation by finding the image of the unit vectors i and j.

Therefore, what I'm recommending is that you find the image of those two vectors

Answer 7

Then those two images will be the columns of your matrix.

To find those two images, you'll need to use your formula.

Answer 8

Make sense?

Answer 9

For example, the matrix of a reflection in the x-axis is
1 0
0 -1

because the i vector remains where it is. The image of i under the transformation is itself.

But the image of j is -j. That's why the second column of the matrix has a negative 1.

Answer 10

:( i fail at linear algebra

Answer 11

I'm struggling to understand because I don't know what a reflection of a matrix is, other than by the definition we were given. Like how do you know what the reflection is about your x-axis without the formula?