Numbered problems are from the text: Linear Algebra with applications, Third Edition, by Otto Bretscher, Pearson Prentice Hall 2005.

Starred problems are challenge problems

**Assignment 1**Due Friday, February 1.- Section 1.1 page 5: 1, 3, 7, 12, 14, 18, 20, 25, 28, 29, 31
- Section 1.2 page 25: 5, 10, 18, 22, 24, 37

**Assignment 2**Due Friday, February 8.- Section 1.3 page 35: 1, 4, 6, 8, 10, 18, 24, 34, 36, 47, 55

**Assignment 3**Due Friday, February 15.- Section 2.1 page 51: 4, 6, 10, 12, 13, 20, 34, 37, 42, 47
- Section 2.2 page 66: 1, 2, 4

**Assignment 4**Due Friday, February 22.- Section 2.2 page 66: 10, 16, 25, 32, 37, 40
- Section 2.3 page 76: 1, 2, 6, 20, 36, 41, 48

**Assignment 5**Due Friday, February 28.- Section 2.2 page 66: 6, 7
- Section 2.4 page 89: 2, 4, 6, 14, 16-25, 44, 48-49, 86
- Section 3.1 page 109: 2, 4, 5, 10, 16, 20, 24, 32, 38, 40, 48

Hint for problem 48 in section 2.4: The line from P_1 to P_3 should be dotted in the figure, being in the back. Note that the plane through P_0, P_1, P_3 is orthogonal to the line spanned by P_2, since P_1-P_0 is orthogonal to P_2 and P_3-P_0 is orthogonal to P_2. Furthermore, the line spanned by P_2 intersects this plane at the center (P_0+P_1+P_3)/3=-(1/3)P_2 of the triangle with vertices P_0, P_1, P_3. Finally, the latter triangle has edges of equal length. Hence the rotation about its center permutes its vertices cyclically. Thus, T permutes the set of vectors P_0, P_1, P_3 cyclically.

**Assignment 6**Due Monday, March 3.- Section 3.2 page 121: 2, 4, 6, 10, 18, 19, 24, 32, 34, 46, 53, 54

**Assignment 7**Due Friday, March 14.- Section 3.3 page 133: 2, 3, 9, 18, 22, 26, 28, 38, 47*, 53, 56*, 60, 61 (starred problems are challenge problems).
- Section 3.4 page 146: 2, 6, 8, 20, 26, 37, 40, 43, 46*, 60, 71

**Assignment 8**Due**Monday, March 31.**- Section 4.1 page 162: 1 to 6, 10, 16, 18, 20, 25, 27, 36, 47, 48, 50, 55

**Assignment 9**Due**Friday, April 4.**- Section 4.2 page 169: 1 to 6, 17, 22, 23, 27, 30, 57 (find a basis for the kernel), 60, 64, 65*, 66,
- Check that xe^{-x} belongs to the kernel of the linear transformation in excercise 40 page 170. Use it and Fact 4.1.7 to find a basis for the kernel of the linear transformation. Carefully justify why the solutions you found are linearly independent, and why they span the kernel.

**Assignment 10**Due**Friday, April 11.**- Section 4.3 page 181: 1, 4, 5, 15, 21, 22, 29, 32, 33, 34, 48, 49

- Find the inverse of the matrix you found in
exetcise 49 page 181. Use it to find the inverse
T^{-1} of the linear trandsformation T
in exetcise 49 page 181, i.e.,
find constants a, b, c, and d, such that

T^{-1}(x cos(t) + y sin(t)) = (ax+cy)cos(t)+(bx+dy)sin(t),

for all scalars x, y.

- Section 4.3 page 181: 1, 4, 5, 15, 21, 22, 29, 32, 33, 34, 48, 49
**Assignment 11**Due**Friday, April 18.**- Section 6.1 page 259: 1, 2, 5, 10, 12, 26, 28, 32, 45, 46, 48, 56
- Section 6.2 page 271: 1, 5, 12, 15, 16

**Assignment 12**Due**Friday, April 25.**- Section 6.2 page 271: 17, 18, 25, 27, 30, *31, 37 (justify all your answers!!!), 38, 46
- Section 6.3 page 287: 1, 2, 3 (translate the triangle
first so that one of its vertices is the origin),

4, 7, 11,

Let A be a 3 by 3 matrix, with det(A)=7, u, v, w three vectors in R^3, such that the parallelopiped determined by them (i.e., the one with vertices 0, u, v, w, u+v, u+w, v+w, u+v+w) has volume 5 units. Find the volume of the parallelopiped determined by Au, Av, Aw. Carefully justify your answer!

**Assignment 13**Due**Monday, May 5.**- Section 7.1 page 303: 1-6, 9, 10, 12, 15 (see Definition 2.2.2 in section 2.2),

Find the matrix of the reflection A of the plane about the line x=y. Find all eigenvalues and eigenvectors of A and a basis of R^2 consisting of eigenvectors of A. Find the matrix of A with respect to the basis you found.

16, 19, 38 - Section 7.2 page 314: 1-4 (see Definition 7.2.6 for the algebraic multiplicity), 8, 12, 14, 15, 17, 19 (see Fact 7.2.8), 22 (use Fact 6.2.7 to write a careful justification), 25, 27, 28*, 29, 33

- Section 7.1 page 303: 1-6, 9, 10, 12, 15 (see Definition 2.2.2 in section 2.2),
**Assignment 13**Due**Friday, May 9.**- Section 7.3 page 325: 1, 2, 7, 8, 9, 10, 12, 19, 21, 22, 24 (Hint: Choose, for example, a matrix similar to the one in problem 23), 27, 28, 36
- Section 7.4 page 338: 1, 2, 5, 11, 12, 13, 22, 25, 27, 47, 52
- Extra Problem (will be graded)

**Assignment 14**Due**Monday, May 12.**- Section 5.1 page 198: 2,15,16,22,26,29
- Extra Problem on diagonalization: (Highly Recommended!!!)

**Assignment 15**Due**Monday, May 21 (Day of the final, not to be handed in, but material will be covered in the final!!!)**- Section 5.2 page 208: 1, 2, 3, 4, 13, 32, 33
- Let W be the plane spanned by the two vectors in
question 5 page 208.
- Find the projection of the vector b=(9,0,9) to W. Note: you will need to first find an orthogonal basis for W. Answer: .5(9, 9, 18)
- Find the distance from b to W. Answer: 4.5 times (square root of 2).