Math 235

Suggested Homework Problems for Brecher's fourth edition, Fall 2014

Note: The homework problems listed here will give the student experience with the ideas and methods of linear algebra. They are not the only such problems; Exams and the homework assigned by your section's instructor may contain questions rather different from these.

Carfully JUSTIFY all your answers to all homework problems. Answers without sufficient justification will not get credit.

Starred problems are challenge problems

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

• 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 (In 37 set up the system of linear equations, find the corresponding augmented matrix. You are not asked to solve it by hand).
• Section 1.3 page 33:   1, 4, 6, 8, 10, 18, 24, 34, 36, 47, 55, 58
• Section 2.1 page 50:  4, 6, 10, 12, *13, 16, 19, 22, 34, 36, 37, 42, 47
• Section 2.2 page 65:  1, 2, 4, 6, 7, 10, 11, 13, 14, 15 (see Example 3), 16, 25, 32, 37, 40, 43.
• Section 2.3 page 77:  2, 4, 6, 14, 17, 18, 20, 29, 30, 38, 41, 50 (Should be CG and GC), 66
• Section 2.4 page 88: 1, 2, 6, 16, 20, 21, 22, 23, 25, 26, 34, 36 (use upper triangular form), 41, 80, 81
Hint for problem 80 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.
• Section 3.1 page 110:  2, 4, 5, 10, 16, 20, 24, 32, 38, 40, 42, 44, 48 (in 48 part c assume both that rank A is 1 and that ker(A) is orthogonal to im(A) and show that A is the projection onto im(A) in the usual sense, ignoring Excercise 2.2.33)
• Section 3.2 page 121:  2, 4, 6, 10, 18, 19, 24, 32, 34, 46, 53, 54
• Section 3.3 page 133:  2, 3, 9, 18, 22, 26, 28, 38, 40, 42, 43, 67*, 73, 76*, 80, 81 (starred problems are challenge problems).
• Section 3.4 page 146:   2, 6, 8, 17. 20, 26, 29, 33, 34, 37 (see hint below), 40 (see hint below), 41 (first interpret this plane as the plane orthogonal to some vector), 43, 46*, 55, 57, 69, 60 (Hint: use the idea of 69), 71
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.
• Hint for problems 33 and 34: 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).
• Section 4.1 page 162:  1 to 6, 10, 16, 18, 20, 25, 27, 36, 47, 48, 50, 55
• Section 4.2 page 170:  1 to 6, 10, 14, 22, 23, 26, 27, 30, 43, 51, 52, 53, 57 (find also 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 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.
• 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, *31, 37 (justify all your answers!!!), 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 289:  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), 16, 19, 38.
• Extra problem for section 7.1: 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 7.2 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*, 29, 33
• Extra Problem for section 7.3
• 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, 41
• Section 7.4 page 340:  1, 2, 5, 11, 12, 13, 16, 22, 25, 27, 47, 52
• Extra Problem on diagonalization (Highly Recommended!!!)
• Section 7.5 page 353: 1, 2, 7, 8, 15, 17, 21, 23, 24.
• Section 5.1 page 199:  2, 15, 16, 22, 26, 29
• Section 5.2 page 209:  1, 2, 3, 4, 13, 32, 33
• Extra problem for section 5.2: Let W be the plane spanned by the two vectors in question 5 page 209.
• 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).
• Extra Problem on projections
• Section 5.3 page 218: 1, 2, 5 to 9, 28 Hint: consider the length squared of L(v+w), 30, 31, 33, 34, 36, 37.
• Section 6.2 page 273:  40 Hint: Recall that det(A)=det(A^T).
• Section 5.4: To be announced