Reflection of a Vector

[Tay]

Ok, Linear Algebra question here. I have no idea what I'm exactly doing wrong, or how everyone else does this. I can't find a good example online that is doing it the way my text book is teaching it.

So here's the problem.

Let L be the line in R^3 that consists of all scalar multiples of
L=

2
1
2

*this is a 1x3 matrix, not sure how to represent a matrix well on here*

Find the reflection of the vector about the line L.
v=

1
1
1

*again, 1x3 matrix*

From what I know, I need to find the orthogonal projection of v onto line L, and then use a handy dandy formula.
I know that a projection is equal to ;
(1/||w||^2)w dot w^T

where ||w||^2 is the magnitude of w squared and w^T is w transpose.

So, I'm saying that w is my line, L. Therefore, ||w||^2 = ((2^2 + 1^2 + 2)^1/2)^2 = ((9)^1/2)^2) = 9

hopefully?

and L dot L^T = 3x3 matrix;

4 2 4
2 1 2
4 2 4

SoOoOOoOo my projection matrix, A, is equal to

(1/9) *

4 2 4
2 1 2
4 2 4

And my handy dandy formula to find the reflection of a matrix with respect to a line is 2proj(x) - x which is also
2Ax - x
(2A - I)x where I is an identity matrix

preforming this out is definitely not getting me the answer in the back of the book though.


I apologize for this really weird way of typing this out, and I really dunno what I'm seeing incorrectly but it's something massive lol.

[Stereo]

The dot product should only produce a scalar, if you have W * W^T then W should be a 1x3 matrix, whereas W^T * W would use the 3x1 matrix. Both of those are W dot W; you transpose the one that makes sense.

Your projection function is just wrong; it should be (a dot b) * b / ||b||^2 to project a onto b. (note: this will produce a vector the same dimension as b, since (a dot b) is a scalar and so is ||b||)

In this case if you were projecting a=(2,1,2)' onto b=(1,1,1)'
so (a dot b) = 5, ||b||^2 = 3
5/3*(1,1,1)'
(not gonna do the actual question for you)


Actually that's another point. ||b||^2 is b dot b.
The 2-norm is sqrt(v_1^2+v_2^2+...+v_n^2), the square root of the sum of squares of each component. So squaring it you get ||v||^2 = v_1^2 + v_2^2 + ...
Which for ||(1,1,1)||^2 = 1^2 + 1^2 + 1^2 = 3 = (1,1,1) dot (1,1,1)

[Tay]

The dot product should only produce a scalar, if you have W * W^T then W should be a 1x3 matrix, whereas W^T * W would use the 3x1 matrix. Both of those are W dot W; you transpose the one that makes sense.

Your projection function is just wrong; it should be (a dot b) * b / ||b||^2 to project a onto b. (note: this will produce a vector the same dimension as b, since (a dot b) is a scalar and so is ||b||)

In this case if you were projecting a=(2,1,2)' onto b=(1,1,1)'
so (a dot b) = 5, ||b||^2 = 3
5/3*(1,1,1)'
(not gonna do the actual question for you)


Actually that's another point. ||b||^2 is b dot b.
The 2-norm is sqrt(v_1^2+v_2^2+...+v_n^2), the square root of the sum of squares of each component. So squaring it you get ||v||^2 = v_1^2 + v_2^2 + ...
Which for ||(1,1,1)||^2 = 1^2 + 1^2 + 1^2 = 3 = (1,1,1) dot (1,1,1)

Wait, but I'm projecting v the (1,1,1) ONTO L which is (2,1,2).

And, I'm not sure I understand your projection equation. In a sense it makes sense compared to my text book. For clarity sake I took a picture of the problem, number 7, and the textbook definitions of the transformations.

Question 7
Orthogonal Projections
Reflections


EDIT: Ok I figured out why I am getting CLOSE to the answer, but not the actual answer. When I am subtracting my identity matrix I am not taking account the scalar being multiplied to A, which is really solving all my issues now.

instead of doing
(1/9)[8 4 8
4 2 4
8 4 8]

minus

(1/9)[9 0 0
0 9 0
0 0 9]
to account for the (1/9) term on the projection matrix I was just subtracting by 1. I feel so pineappleing dumb. Lordddddd, thank you @Stereo for the help!

[Stereo]

Yeah, the projection to focus on is proj_L(x) = (x dot w / w dot w) w

Which ends up being what I said when you move the terms around a bit (and note that ||w||^2 = w dot w)