# Suggested Problems

### (Kolman, 6th edition)

This is **not** a homework. I do not expect you to solve all these problems;
just make sure that you **know** how to solve them, and you'll do well
in the exams. Do **not** try to write down all solutions; most problems
are solved in two words provided that you understand the subject. (That,
of course, doesn't apply to those where you actually need to calculate
something; solve these ones till you feel confident.)
**Matrices.** Your principal goal here is to feel confident with
matrix operations (including inverse)
and relation between matrices and linear systems
and be able to convert a matrix to its reduced row echelon form/solve
a linear system.

**1.2:** 1-8, 14, 20, 22
[p.16] [p.17]
**1.3:** 20, 29
[p.25] [p.26]
**1.4:** 1-16
[p.35] [p.36]
**1.5:** 1-14
[p.54] [p.55]
**1.6:** 8, 10, 16
[p.63]
**Vector spaces.** You should understand the concept of vector space and subspace,
to be able to detect a subspace, and be confident with the spaces like
**R**^{n}, **R**_{n},
the space of matrices, and
the space *P*_{d} of polynomials. Besides, for now
(**2.2-2.4**), you should be able to solve the following problems:
(1) does a given vector belong to a given span?
(2) do given vectors span the space?
(3) are given vectors linearly independent?
(4) find a basis in a given span;
(5) complete a given linearly independent set to a basis;
(6) find a basis in the solution space of a system.
Note that all problems reduce to linear
systems, which you should be able to write (for any of the spaces above),
investigate, and, most importantly, interpret the result.

**2.2:** 1-20 (mainly, understanding of the concepts)
[p.98]
**2.3:** 1-22 (understanding), 23-26 (calculations)
[p.107] [p.108] [p.109]
**2.4:** 1-13 (calculations), 18-28 (understanding)
[p.119] [p.120]
**2.5:** 1, 2, 4-6, 11-14, 26, 27, 29-31 (calculations),
16, 17, 21, 22, 35, 36 (understanding)
[p.134] [p.135] [p.136]
**2.6:** 1-16, 21, 22 (calculations), 37-41 (understanding)
[p.150] [p.151] [p.152]
**2.7:** 1-8, 11-16 (calculations), 20 (understanding)
[p.159] [p.160]

[**Midterm II**]

**2.8:** 1-8, 12-13 (calculations), 28-42 (understanding)
[p.169] [p.170]
[p.171] [p.172]
**Inner product spaces.** The concept of an inner product/inner product
space; typical inner products (standard inner product in **R**^{n},
definite integral inner products in functional spaces); matrix of an
inner product (for small matrices, detecting whether it is positive definite
by completing a square); norm (length), angle, and distance;
orthogonal and orthonormal bases, Gram-Schmidt process;
orthogonal complement, orthogonal projection, distance to a subspace, approximation.

**3.3:** 6, 7, 16-21, 38, 42-49 (understanding),
29, 30, 33, 34 (calculation)
[p.205] [p.206] [p.207]
**3.4:** 11-18, 24, 25 (calculation)
[p.217] [p.218]
**3.5:** 2-8, 11, 12, 14, 15, 16, 19, 20 (calculation),
23, 25, 26 (understanding)
[p.231] [p.232]
**Linear transformations.** The concept and examples; kernel and range; matrix of a
linear transformation; similarity.

**4.1:** 1-3, 7, 9-22, 28 (understanding)
[p.251] [p.252]
**4.2:** 5-12 (calculation), 15, 18, 20, 24-26 (understanding)
[p.264] [p.265]
[p.266]
**4.3:** 1, 2, 6, 11, 18, 19 (calculation)
[p.274] [p.275]
[p.276]

[**Final**]

**4.6:** 5, 9, 11 (calculation), 6-8, 10 (understanding)
[p.297]
**4.suppl:** 16, 18, 19 (understanding)
[p.299]
**Determinants.** The concept, properties, evaluation, and
applications.

**5.2:** 1, 2 (calculation),
3, 4, 8-12, 7, 14-20, 27 (understanding)
**5.3:** 11-13, 15, 16 (understanding)
**Diagonalization.** Eigenvalues, eigenvectors, characteristic
polynomial, diagonalizability and diagonalization. Diagonalizability
of symmetric matrices (self-adjoint operators); orthogonal matrices and
operators.

**6.1:** 3, 5, 7, 8, 16 (calculation),
4, 6, 12, 14, 19, 20, 25, 29 (understanding)
**6.2:** 15-20 (calculation),
30, 33-35 (understanding)

*Last update: December 07, 2004 *