- #1
negation
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- 0
Homework Statement
Given below are three geometrically defined linear transformations from R3 to R3. You are asked to find the standard matrices of these linear transformations, and to find the images of some points or sets of points.
a) T1 reflects through the yz-plane
b) T2 projects orthogonally onto the xy-plane
c) T3 rotates clockwise through an angle of 3π/4 radians about the z axis
The Attempt at a Solution
a)
The standard matrix of T1 is
-1,0,0
0,1,0
0,0,1
b)
The standard matrix of T2 is
1,0,0
0,1,0
0,0,0
c)
The standard matrix of T3 is
-sqrt(2)/2, sqrt(2)/2,0
-sqrt(2)/2,-sqrt(2)/2,0
0,0,1
d)
The image under T1 of the line segment joining the points (-2, -4, 3) and (2, 2, 4) is line segment joining the points
(2,-4,3) and (-2,2,4)
e)
The point (-4, -4, -4) is first mapped by T2 and then T3. The coordinates of the resulting point are
[1,0,0;0,1,0;0,0,0] [-sqrt(2)/2, sqrt(2)/2,0; -sqrt(2)/2,-sqrt(2)/2,0; 0,0,1] =
[-sqrt(2)/2,0,0; 0,-sqrt(2)/2,0; 0,0,0]
[-sqrt(2)/2,0,0; 0,-sqrt(2)/2,0; 0,0,0] [-sqrt(2),0,0;0,-sqrt(2)/2,0;0,0,0]
=[0.5,0,0;0,0.5,0;0,0,0]
(e) is wrong but why?