CPSC 453 Tutorials Xin Liu Oct 16, 2013

HW1 review Why I was wrong?

Q1 Determine the affine transformation on the plane taking the triangle with vertices (1, 1) (1, 2), and (3, 3) to the equilateral triangle with with vertices (1, 0), (-1, 0), and (0, sqrt(3)) Solution: Refer to a Linear Algebra textbook for Inverse Matrix calculation

Q2 Let P, Q, R be points on the 2D affine plane. Show that for an arbitrary scalar, is a point but is a vector Solution: for any point The last component of X is 1, because Therefore, X is a point. The last component of Y is 0, because Therefore, Y is a vector.

Q3 Define what it means for a transformation in R n to preserve angles. (a) Show that an isometry preserves angles. (b) Give an example of a transformation that preserves angles but is not an isometry. Solution: Let T be a linear transformation in R n. T is angle preserving iff An isometry transform, L(u) is an orthogonal transformation

Q4 Let the frame F on the plane be obtained from the cartesian reference frame by a counter-clockwise rotation about the origin through 135 degrees. Find the transfer matrices. An ellipse has equation 5x 2 +6xy+ty 2 =1 in cartesian; what is its equation in the frame F?

Q5 Find the transformation matrix for a rotation by a 120-degree angle about the axis defined By the unit vector r = 1/sqrt(3)(1, 1, 1). (This of course can be done using the result of the previous exercise, but you might be able to guess the matrix directly by considering what the transformation does to the unit cube [0, 1] 3. A permutation of axis: x->y, y->z, z->x