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Question 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, clarifications, or extra instructions.
Be sure to follow these! Note: Only a selection of exercises may be
graded. 1) A reflection 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 reflection across L. L
a. What happens to any 56 if you apply the reflection ‗- 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 reflection across one line followed by a reflection across
another line always ends up being a rotation. This can be proven with
matrix multiplication, as follows. The matrix for a reflection 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 reflection across the line with angle a with
another reflection 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 find a basis for the subspace (with
some explanation of how you know it is a basis) and the dimension of
the subspace
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