Question

How to find the standard matrix for a rotation that has the same effect as the reflection H_(pi/3) followed by H_(pi/6) if they are standard matrices for the reflections of R^2 about lines through the origin making angles with positive x axis?

Answer 1

I used the form [cos2theta sin2theta; sin2thea -cos2theta]

Answer 2

and I thought you just need to sub in 60 and 30 degrees into it and then multiply them out?

Answer 3

That sounds correct. Im a little lost on your notation though.
What does
\[H_\theta\]
stand for?

Answer 4

i know its going to be something completely obvious lol >.<

Answer 5

It reflects a vector x about a line through the origin that makes an angle theta with the positive x axis

Answer 6

so standard matrix for reflection is denoted as H_theta in my text

Answer 7

ah, gotcha, alright let me think about this for one sec....

Answer 8

okay well I keep getting it wrong and I tried [cos 60 sin 60; sin 60 -cos60]*[cos 120 sin 120; sin 120 -cos120]

Answer 9

a book i have has the reflection matrix as:

2c^2 - 1 2cs
2cs 2s^2-1
Where c is cos, and s is sin.

Answer 10

Well in my book it states H_theta x = [cos 2theta sin 2theta; sin2theta -cos2theta][x;y]

Answer 11

what does the s stand for?

Answer 12

oh hahaha never just saw it

Answer 13

ima scan the section in my book, one sec.

Answer 14

Oh snap I got it

Answer 15

hm that's strange but yeah mine only works for things in R^2

Answer 16

thanks for your help regardless btw do you know much about elementary matrices?

Answer 17

what did you get as a final answer? im doing it with the formula in my book just to see how this works.

Answer 18

and yes i know a smidge about Elementary Matrices.

Answer 19

[0.5 sqrt(3)/2; -sqrt(3)/2 0.5]

Answer 20

Do you know how to find the product of elementary matrices like [0 -2; 4 0]? I can't seem to get them

Answer 21

would you row reduce and record each step

Answer 22

cool our fomulas both work

Answer 23

which each row operation, you create the elementary matrix that corresponds to it. The elem. matrix is like a record of what youve done. i dont know if there's a fast way to find the product of elem. matrices though. Do you have an example you would like to work out?

Answer 24

[0 -2; 4 0]

Answer 25

We are going to row reduce that right?

Answer 26

yeah I just got stuck when you had to switch the rows

Answer 27

so to swap rows, you would multiply that matrix (on the left) by:
0 1
1 0

Answer 28

actually okay I'm just lost in general actually can you do [3 4; 2 1]

Answer 29

[3 4; 1 2]** my bad

Answer 30

sure, i'll do it on paper though, one sec >.<

Answer 31

in that product notation in the last pic, it should be:
\[\prod_{i=1}^{4}E_{5-i}\]

Answer 32

what's that symbol??

Answer 33

See I'm so lost b/c how'd you know that the -1/2 for E_3 would go there if you -1/2R to the second row how'd you know it doesn't go to the bottom left spot

Answer 34

its like the summation, but product instead.
\[\prod_{i=1}^{4}E_{5-i} = E_{5-1}*E_{5-2}*E_{5-3}*E_{5-4} = E_4*E_3*E_2*E_1\]

Answer 35

If i had put it in the bottom left spot, it would effect the first row and not the second. If the bottom row of my elem matrix had been [-1/2, 1], thats me saying " i want negative 1/2 of the first row plus the second row"

Answer 36

i only want to divide the second row by 2, so the elementary matrix has to be:
1 0
0 -1/2
This is saying , "I dont want any of the first row, and multiply the second row by -1/2

Answer 37

oh okay so you'd put it with the row you're multiplying it with and the column you want it to be affected by?

Answer 38

yes. Theres a good lecture online if you want more info on it, let me see if i can find it...

Answer 39

http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-2-elimination-with-matrices/

Answer 40

skip to about 24:30

Answer 41

oh okay thanks for all your help!! wish I could give you more medals I'll check the link out

Answer 42

no prob :) it was an interesting problem!