Homework Assignments for Math 235 Section 4
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 Thursday, September 4.
 Section 1.1 page 5: 1, 3, 7, 12, 14, 18, 20, 25, 28, 29, 31
 Assignment 2
Due Thursday, September 11.
 Section 1.2 page 25: 5, 10, 18, 22, 24, 37
 Section 1.3 page 35: 1, 4, 6, 8, 10, 18, 24, 34, 36, 47, 55
 Assignment 3
Due Thursday, September 18.
 Section 2.1 page 51: 4, 6, 10, 12, *13, 20, 34, 36, 37, 42, 47
 Assignment 4
Due Thursday, September 25.
 Section 2.2 page 66: 1, 2, 4, 6, 7, 10, 11, 13, 16, 25, 32, 37, 40, 43
 Section 2.3 page 76: 21, 22, 23, 25, 26
 Assignment 5
Due Thursday, October 2.
 Section 2.4 page 89: 2, 4, 6, 14, 1625, 44, *48, 49, 86
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_1P_0 is orthogonal to P_2 and P_3P_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 109: 14, 15, 16, 19, 20, 21
 Assignment 6
Due Tuesday, October 7 (date of midterm 1).
 Section 3.1 page 109: 2, 4, 5, 10, 16, 20, 24, 32, 38, 40, 48
 Section 3.2 page 121: 2, 4, 6
 Assignment 7
Due Thursday, October 16.
 Section 3.2 page 121: 10, 18, 19, 24, 32, 34, 46, 53, 54
 Assignment 8
Due Thursday, October 23.
 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, 17.
 Assignment 9
Due Thursday, October 30.
 Section 3.4 page 146: 20, 26, 29, 33, 35, 37, 40, 41, 43, 46*, 69, 60 (Hint: use the idea of 69), 71
 Note that new problems were added above.

Carfully JUSTIFY all your answers to all homework problems.
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).
 Assignment 10
Due Thursday, November 6.
 Section 4.1 page 162: 1 to 6, 10, 16, 18, 20, 25, 27, 36, 47, 48, 50, 55
 Section 4.2 page 169: 1 to 6, 23, 51, 52.
Note: In questions 1 to 6 and 23 just determine if the transformation is linear. You do not need to determine if it is an isomorphism,
as we did not define this notion in class yet.
Carfully write down the verification of the two conditions of linearity, or
show that at least one of them is violated.
 Assignment 11
Due Thursday, November 13.
 Section 4.2 page 169: 7, 10, 14, 22, 26, 27, 30, 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 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 12
Due Tuesday, November 18.
 Section 6.1 page 259: 1, 2, 5, 10, 12, 16, 26, 28, 32,
36 (see Example 5), 45,
46, 48, 56
 Assignment 13
Due Tuesday, November 25.
 Section 6.2 page 271: 1, 5, 12, 15, 16, 30, *31,
37 (justify all your answers!!!), 38, 46
 Let V be an ndimensional 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 ith column, by definition, is b_i=Be_i=[T(S(e_i))]=[T(v_i)].
Thus, the ith 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 271:
17, 20
 Assignment 14
Due Thursday, December 4.
 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!
 Section 7.1 page 303:
16, 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:
14 (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 15
Due Thursday, December 11.
 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)

Extra Problem on diagonalization: (Highly Recommended!!!)
 Section 5.1 page 198: 2,15
 Assignment 16
Due Thursday, December 18 (Day of the final. Not to be handed in,
but material is covered in the final exam).
 Section 5.1 page 198: 16,22,26,29
 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).