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## MAT 211: Homeworks

Homework is a very important and mandatory part of the course. Homework assignments are due each week at the beginning of the Wednesday class (12:00am). Under no circumstances will late homework be accepted.

You are encouraged to discuss homework with other students in class, however the work you hand in should be your own. Homework that appears to be copied from someone else will receive a grade of 0 and may result in charges of academic dishonesty.

Each homework will consist of two parts: practice problems (P) and problems to hand in (THI) .
Practice Problems: These are not to be handed but are essential to master the course. Similar problems may appear on the Midterm Exams and Final. These will be designated (P) in the assignments below.
Problems to hand in: These should be done after you’ve mastered the practice problems. Your solutions must be carefully and legibly written and handed in in class on the due date. These will be designated as (THI). These will be graded.

HW Due date Assignment Remarks
HW 1 W 9/2 (P): p.5 questions 3,7,9,11,13,24,25,31,33,48; p.18 questions 1,3,5,11,13,31,39
(THI): p.5 questions 2,4,8,12; p.18 questions 4, 6, 8, 32, 48
Please check solutions of linear systems by plugging in!
HW 2 W 9/9 (P) p.34 questions 1, 3, 5, 9, 10, 11, 13, 17, 19, 23, 25;
p.53 questions: 4, 5, 7, 25, 35
(THI) p.34 questions 2, 6, 16, 18, 24;
p.53 questions 6, 10, 20, 22, 32
1.3 Rank of a matrix is related to the number of solutions of the system.
2.1 Learn what Linear transformations are and how to visualize them!
HW 3 W 9/16 (P) p.71 questions 9, 11, 19, 25, 26ab; and also questions 34 on p.36, 49 p.37, 31 p.54, 35 p.55
(THI) p. 71 questions 2, 6, 8, 10, 12, 20, and also 36 on p.36
We are focusing on the geometry of linear transformations, and how to write and interpret the corresponding matrices. Please solve the questions about projections and reflections from scratch (do not just plug in into the formulas in 2.2). Follow the strategy discussed in class on 9/9 (we'll continue with this material on 9/11).
HW 4 W 9/23 (P) p.85 questions 1,3,7,9,35, 49; p.97 questions 1, 3, 9, 10, 11, 29, 33, 41 ; also 12 on p.54
(THI) p. 85 questions 2, 4, 10, 12, 40, 50; p. 97 questions 2, 4, 58
Matrix multiplication is a very important operation. Make sure you know both the computation recipe and its meaning as composition of transformations. Pay attention to the order of operations: when applying AB to a vector x, you apply B first, A second: ABx = A(B(x)).
HW 5 W 9/30 (P) p.119 questions 4,6, 15, 16, 17,19,33;
p.131 questions 11, 18, 19, 20, 27, 29, 47
(THI) p. 119 questions 8, 12, 22, 30; p.131 questions 10, 14, 16, 24, 32
Kernel and image are important examples of subspaces. Linear (in)dependence and basis are cornerstone notions of linear algebra!
HW 6 W 10/7 (P) p.143 questions 11, 13, 15, 36, 38; p.159 questions 19, 43, 47, 48, 49
(THI) p. 143 questions 6, 8, 20, 28, 30; p.159 questions 12,18,22, 26,44
3.3. It is important to be able to detect linear (in)dependence, not only by solving for relation, but also "by inspection" in simple cases. Practice with kernel and image (ker and im tell you a lot about solutions to the linear system).
Hint for #30, p.143: you can think of this subspace as kernel of something...
3.4. Changing the basis often simplifies things quite a lot. It is worth the effort finding the matrix of a transformation in new coordinates!
HW 7 F 10/23 (P) p.176 questions 5, 33, 40, 47; p.184 questions 11, 13, 51, 63, 64
(THI) p. 176 questions 2, 26, 30, 48; p.184 questions 6, 12, 22, 54, 60, 62
We haven't yet discussed the notion of isomorphism in class, so the homework will be accepted until Friday, 10/23. You are encouraged to read 4.2.
Chapter 4: a vector space can consist of functions, matrices, whatever else instead of familiar vectors... but "the rules" remain the same, and you can still have subspaces, linear transformations, and the rest of linear algebra. Pretty much everything hinges on making linear combinations using sums and scalar multiples.
HW 8 W 10/28 (P) p.195 questions 1, 23, 25, 29, 51; p.214 questions 3, 11
(THI) p. 195 questions 2, 4, 8, 10, 14, 24; p.214 questions 2, 6, 10
Section 5.1 will be discussed in class on Monday. In the meanwhile, read the book! The homework only concerns a few easy basics about angles.
In 4.3, we are only covering the part that deals with writing a matrix of a linear transformation in a given basis. Make sure to stick to particular basis for every step of your solution!
HW 9 W 11/4 (P) p.216 question 15, p. 224 questions 3, 9, 33, 39; p.233 questions 1, 2, 3, 4, 37
(THI) p. 217 question 26; p. 224 questions 4, 6, 14, 29, 32, 34; p.233 question 36
We omit QR factorization in 5.2.
5.1-5.2 Orthonormal bases are very convenient to work with (for example, if you are concerned with projections), so we are putting some effort into finding them. Orthonormal transformations (section 5.3) preserve lengths and angles!
HW 10 W 11/11 (P) p.260 questions 1, 7, 8; p. 275 questions 3, 5, 7, 17, 31, 39; p.289 question 5
(THI) p. 260 questions 9, 10; p. 275 questions 4, 8, 14, 18, 36, 40, 42; p.289 questions 4, 16
Questions 4, 5 on p.289 use elimination process to compute determinant. Note that you don't have to reduce all the way to rref - a diagonal or triangular matrix is fine, and so is a matrix with a lot of zeroes. Make sure to track all the swaps and signs.
5.5 You can have a "dot product" (inner product) in a function space!
6.1-6.2 Determinants may be a bit of a pain to define, but they have nice properties and detect whether the matrix is invertible.
HW 11 W 11/18 (THI) p. 305 questions 1, 2, 3, 22, 24

Homework is shorter this week: review the rest of material for the test on 11/18
Going over the past homeworks (including practice questions!) is a must. End-of-chapter questions provide a good concept check.
6.3 Geometric meaning of determinants (areas, volumes); Cramer's rule
HW 12 W 12/2 (P) p.323 questions 4, 9, 13; p. 336 questions 2, 22, 45;
p.345 questions 2,11, 13, 23
(THI) p. 323 questions 2, 60; p. 336 questions 4, 8, 10;
p. 345 questions 4, 8, 14, 18, 42