# Homework Assignments for Math 235 Section 3

Numbered problems are from the text: Linear Algebra with applications, Fourth Edition, by Otto Bretscher, Pearson Prentice Hall 2009.

• Assignment 1   Due Tuesday, September 14.
• Section 1.1 page 5:   1, 3, 7, 12, 14, 18, 20, 25, 28, 29, 31
• Section 1.2 page 18:   5, 10, 18, 22, 24, 37
• Assignment 2   Due Tuesday, September 28.
• On line WebWork HW3 covers sections 2.3 and 2.4 and will be due after the following handwritten homework assignment. Unfortunately, the common WebWork assignments do not cover sections 2.1 and 2.2 of the text, although the material in these sections is part of the syllabus. Hence, additional hand written problems are assigned to section 2.1. My appologies for the confusion.
• Section 2.1 page 50:   4, 6, 10, 12, 16, 19, 22, 34, 36, 37, 42, 47.
• Section 2.2 page 65:   1, 2.
• Assignment 3   Due Tuesday, October 5.
• Section 2.2 page 65:   4, 6, 7, 10, 11, 13, 14, 15 (see Example 3), 16, 25, 32, 37, 40, 43.
• Section 2.4 page 88:   21, 22, 23, 25, 26
• Note: Bretcher interchanged the order of sections 2.3 and 2.4 from the third to the fourth edition of his text. We cover section 2.4 in class (The inverse of a linear transformation) before section 2.3 (Matrix products).
• Assignment 4   Due Thursday, October 14.
• Section 2.3 page 77: 29, 30, 38, 41, 50 (Should be CG and GC)
• Section 2.4 page 88: 36, 41
• Assignment 5   Due Thursday, October 21.
• Section 3.1 page 110:  24, 32, 38, 40, 48
• Assignment 6   Due Thursday, October 28.
• Section 3.2 page 121:  34
• Section 3.3 page 133:  2, 3, 9, 18, 22, 26, 28, 38, 39, 80
• Assignment 7   Due Thursday, November 4.
• Section 3.4 page 146:   2, 6, 8, 17. 20, 26, 29, 33, 35 (omit the geometric interpretation for this problem), 37, 40, 41 (first interpret this plane as the plane orthogonal to some vector), 43, 55, 57, 69, 60 (Hint: use the idea of 69), 71
• Hint for problems 33 and 35: Assume only that v_1, v_2, v_3 are three unit vectors that are pairwise orthogonal (i.e., v_i dot v_j is zero, if i is different from j). You do not need to use vector product here. It follows that {v_1, v_2, v_3} is a basis for R^3 (you may assume this).
• Hint for problems 37, 40, and 41: Guess a basis related to the geometric problem and check that the matrix is diagonal by computing the matrix.
• Assignment 8   Due Wednesday, November 10 (a Thursday schedule).
• Section 4.1 page 162:  1 to 6, 10, 16, 18, 20, 25, 27, 36, 47, 48, 50, 55
• Assignment 9   Due Thursday, November 18.
• Section 4.2 page 170:  1 to 6, 10, 14, 22, 23, 26, 27, 30, 51, 52, 53, 43,57 (find also a basis for the kernel), 60, 64, 66,
• Check that xe^{-x} belongs to the kernel of the linear transformation in excercise 40 page 171. Use it and Theorem 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 Tuesday, November 23.
• Section 4.3 page 181:  5, 15, 21, 22, 29, 32, 33, 34, 48, 49
• Section 6.1 page 259:  1, 2, 5, 10, 12, 16, 26, 28, 32, 36, 45, 46, 48, 56
• Section 6.2 page 273:  1, 5, 12, 15, 16, 30
• Assignment 11   Due Thursday, December 2.
• Section 6.2 page 273:  37 (justify all your answer!!!), 38, 46
• Let V be an n-dimensional vector space with basis {v_1, ..., v_n}, [ ]:V -> R^n the coordinate linear transformation, and S:R^n->V its inverse, given by S(c_1, ... c_n)=c_1v_1+ ... +c_nv_n. Let T:V->V be a linear transformation. We get the composite linear transformation from R^n to R^n, mapping a vector x to [T(S(x))], i.e., to the coordinate vector in R^n of the vector T(S(x)) in V. Being linear, the above transformation is given by multiplication by a square n by n matrix B, i.e., [T(S(x))]=Bx, for all x in R^n. The matrix B is called the matrix of T in the given basis (section 4.3). Its i-th column, by definition, is b_i=Be_i=[T(S(e_i))]=[T(v_i)]. Thus, the i-th column of B is the coordinate vector of T(v_i). The determinant det(T) is defined to be det(B) (Def 6.2.11). Use the equation b_i=[T(v_i)] and the standard basis {1, x, x^2} of P_2 and the standard basis of R^{2 x 2} to solve the following problems in section 6.2 page 273:
17, 20
• Section 6.3 page 273:  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!
• Section 7.1 page 305:  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.
• Section 7.1 page 305:  16, 19, 38.
• Assignment 12   Due Thursday, December 9.
• Section 7.2 page 317:  1-4 (see Definition 7.2.6 for the algebraic multiplicity), 8, 12, 14, 15, 17, 19 (see Fact 7.2.8), 22 (use Theorem 6.2.1 to write a careful justification), 25, 27, 28 (a challenge problem), 29, 33
• Extra Problem
• Section 7.3 page 327:  1, 2, 7, 8, 9, 10, 12, 13, 16, 19, 21, 22, 24 (Hint: Theorem 7.3.6 part c suggests that we choose a matrix similar to the one in problem 23), 27, 28 (see Definitions 7.2.6 and 7.3.2), 36.
• Section 7.4 page 340:  1, 2, 5, 11, 12, 13, 16, 22, 25, 27, 47, 52.
• Assignment 13   Due Thursday, December 13 (day of final, not to be handed in, but material will be covered in the final).
• Section 7.5 page 353: 1, 2, 7, 15, 17.