numbers refer to the exercises at the end of each section of the textbook (1St edition numbers are given if they are different). You must show your work to get full marks. For some questions, I have given hints, clarications, or extra instructions. Be sure to follow these! Note: Only a selection of exercises may be graded. 1) A reection in R2 across a line I. through the origin is a linear transformation that takes a vector 56 to its \"mirror image\" on the opposite side of L. Suppose that M is the standard matrix for reection across L. L a. What happens to any 56 if you apply the reection '- g} transformation twice? \\ b. Explain why this tells you that M 2 = I, must be true. a \\ \\ 3 c. Show that it follows that M '1 = M . d. Bonus: It is a remarkable fact that doing a reection across one line followed by a reection across another line always ends up being a rotation. This can be proven with matrix multiplication, as follows. The matrix for a reection across the line that is inclined at angle a is given _ cos(2a) sin(2a) by M\" ' sin(2a) -cos(2a) ' and the matrix for a rotation by angle 6' is given by R9 = sin 6 cos 6 cos!) sing ]_ Use matrix multiplication and trigonometric identities to prove that the composition of a reection across the line with angle a with another reection across the line with angle 5 is a rotation. What is the angle of this rotation? 2) Do textbook question 3.3 #54. a 3) Let S be the subset of R4 consisting of all vectors that have the form 3; 3g , b where a and b are any scalars. a. Find four different vectors that are in S. You should show how you know they are in S. Then write down one vector in R4 that is not in S, with an explanation of how you know that. b. Prove that S is a subspace of R4 by proving that it satisfies all three conditions of a subspace. Then nd a basis for the subspace (with some explanation of how you know it is a basis) and the dimension of the subspace. \f