Find the standard matrix of the linear transformation from R2 to R2 that first reflects though the
x1-axis and then reflects through the line x1=x2
Solution
Lets think what both reflection does this one is simple the vector (x,y) goes to the vector (x,-y)
now reflection across the line x1=x2 take the vector (1,1) as the direction of the line. take the
orthogonal projection of (x,y) to this vector that one is
(x,y) - ((x+y)/2 , x+y/2) = ((x-y)/2, (y-x)/2)).
now reflection across that line is (x,y)- 2((x-y)/2, (y-x)/2)). If you do the picture it will be clear
all you are doing is moving along the line perpendicular to the line x1=x2 that goes through (x,y)
and you move twice the distance from (x,y) to that line
So reflection across the line x1=x2 sends the vector (x,y) to (y,x)
Thus combining both reflections we have
(x,y)->(x,-y)->( -y,x)
To obtain the standard matrix find what happens to the vector (1,0)
(1,0)-> (0,1)
and the vector (0,1)
(0,1)-> (-1,0)
And the standard matrix is
T=
[0..-1]
[1.. 0]
all that is is putting the result of (1,0) as the first column and the result of (0,1) as the second
column..
![Find the two nearby machine numbers x- and x+
Show all steps and working.
Here is an example for x=3/10 answer question in this form:
Solution
x- = (1.00110011001100110011001)2 * 2-2 x- = [2 + 2^-1 + 2^-3 + 2^-4 + 2^-7 +
2^-8 + 2^-11 + 2^-12 + 2^-15 + 2^-16 + 2^-19 + 2^-20 + 2^-23]*2-2 x- = [1/2 + 2^-3 + 2^-5 +
2^-6 + 2^-9 + 2^-10 + 2^-13 + 2^-14 + 2^-17 + 2^-18 + 2^-21 + 2^-22 + 2^-25] x- = [0.5 + 0.125
+ 0.03125 + 0.015625+...............] x- = 27/40 = 4/7(approx) similarly x+ = 27/40 = 4/7](https://image.slidesharecdn.com/findthetwonearbymachinenumbersx-andxshowallstepsandwo-230402083112-9591a42f/85/Find-the-two-nearby-machine-numbers-x-and-x-Show-all-steps-and-wo-pdf-1-320.jpg)
