# Homework Assignments for Math 235 Section 1

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.
• 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
• 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.
• 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).