** 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