# 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 Rn, Rn, the space of matrices, and the space Pd 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 Rn, 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