Math 223, Section 101

Homework
There will be 10 homework assignments. Homework is due at 10:00,
before class begins. No late homework will be accepted.

Your two lowest homework scores will be dropped. No further concession
will be made, for any (including medical) reason.

Homework sets not picked up in class are in a filing cabinet
outside Math Annex 1213.

Solutions
Note: Most of the solutions are taken from homework handed in by you.
If you recognize your solutions, and object to them being posted on
the internet, please sent an email to the instructor, they will be
removed. Names have been removed.
Homework 1. Due Wednesday, September 16.   Do the following problems from the textbook.

1.1.1 (b), (d), (e)1, (g)2
1.1.2 (a)3, (b), (c)
1.1.3 (b)
1.2.1 (c), (d), (e), (f)
1.2.2. (a), (b), (d)

Hints:
1 First try to do the case that O=A. Go backwards: i.e., construct the point 1/3(OA+OB+OC), and then explain why that point is the barycentre.
2 Describe a method for the construction, you don't actually have to do the construction.
3 To prove the triangle inequality, you may try to start with the triangle inequality, take the square, and then try to reduce it to the Cauchy-Schwartz inequality.
Homework 2. Due Wednesday, September 23.   Do the following problems from the textbook.

1.1.3 (c). If it makes it easier for you, just treat the case where 0 < a < c and 0 < b < d.
1.2.2 (f), (g), (h). For (g), use the properties of ellipses and hyperbolas given in (d) and (e)
1.2.3 (a), (c), (e).
In (c) you need to sketch the surface in 3-space with equation z=x ² ± y ². Start with horizontal cross sections z=constant, or intersections with the coordinate planes, x=0, y=0, z=0.
For (e), notice that rescaling the second coordinate by a factor of 2 will change the shape of the curve, but not whether it is an ellipse or a hyperbola. Once you rescale the second coordinate, you are in Case (a).
1.3.1 (b), (c), (d)
Homework 3. Due Wednesday, September 30.   Do the following problems from the textbook.

1.3.1 (e). The problem does not say that the transformations fix the vector e1, it only says that the vertex A of the triangle is at the head of e1. Note that 131b solves the same problem for the square centred at the origin with vertices at the unit points on the axes.
1.3.2 (c), (d), (e), (f). The notation R0 stands for the reflection across the x-axis (see Page 14). The geometric fact underlying both (c) and (f) is that the successive reflection across two lines is equal to rotation by twice the angle between the two lines. If you manage to prove this geometrically, you can try to adjust your proof to the cases required under (c).
1.3.3 (b), (e), (f). If you don't know the meaning of range, look it up here. For example, projection onto the x axis has the x-axis as range.
1.3.4 (c), (f), (h)
Homework 4. Due Wednesday, October 14.  

Do Problem 1.5.3(e) from the textbook.

Do the following problems from the notes on dynamical systems posted under additional course material:

1.3, 1.4, 1.5, 1.6, 1.7, 1.8
Homework 5. Due Wednesday, October 21.   Do the following problems from the textbook.

1.4.3 (a), (b)
1.5.1 (b). Compute complex eigenvalues and eigenvectors for the four off-diagonal matrices (0,1,1,0), (0,-1,1,0), (0,1,-1,0), (0,-1,-1,0).
1.5.2 (b), (c)
1.5.3 (c), (d), (f). For (c), diagonalize the matrix using complex numbers and write a formula for the n-th power of the matrix.
Sample Midterm Exam 3. (Page 34 of the textbook.)
Homework 6. Due Wednesday, October 28.   Do the following problems from the textbook.

3.1 (h)
3.2.1 (a), (i)
3.2.2 (b), (c), (d), (e)
3.2.3 (a), (b), (c), (d), (e)
Homework 7. Due Friday, November 13.   Do the following problems from the textbook.

3.3.1 (a), (b). For (b), also find the volume.
3.3.2 (b), (c)
Homework 8. Due Wednesday, November 18.  

3.3.3 (a), (b), (c), (e), (f), (g), (h)
3.3.4 (c), (e), (f)
From the notes on dynamical systems, do 1.2 and 1.10.
Homework 9. Due Friday, November 27.  

3.4.2 (a), (c), (e)
3.5.1 (a) the last two only, (b), (g), (h)
3.5.4 (a), (b) the last two only, (c)
Homework 10. Due Friday, December 4.  

3.5.3 (a), (b), (d)
In (b) it refers to the description of orthogonal transformations as either (A) (orientation preserving case) a rotation about some axis through the origin (B) (orientation reversing case) a rotation about some axis through the origin, composed with the reflection across the plane orthogonal to the rotation axis. For each transformation (i), (ii), (iii), (iv), say if it is in Case A or B, and try do say as much as possible about the rotation axis and the rotation angle.
In (d) it should read: Find orthonormal bases in these orthogonal complements (one basis for each complement).
3.6.1 (c), (d), (f), (i), (k).
When it says to diagonalize a matrix, this means find the change of basis matrix and the diagonal matrix, but there is no need to compute the inverse of the change of basis matrix.
3.6.2 (b) (c), (e), (f).
In each case find both the general solution and the solution of the given initial value problem.
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