HW #

Homework problems

Due date



Cumulative Final Exam on Tuesday, May 10th, 7:309:30am, usual classroom.
One page two sides of reference sheet is allowed (as usual, no problems solutions)
Training set of problems (don't forget about ALL exams, ALL quizzes(!); other problems from HWs and examples in the text might be helpful):
0) One.III.1, p. 54: 1.12 (a,b,d); One.III.2, p. 62: 2.27;
1) Two.III.1, p. 118: 1.18 (a,b), 1.19 (a,b), 1.20 (a,b);
2) Three.I.1, p. 172: 1.12 (b), 1.15, 1.16, 1.17 (a,c,d);
3) Three.II.2, p. 200: 2.23 (a), 2.27;
4) Three.IV.4, p. 252: 4.15 (a,c,d), 4.16;
5) Three.V.1, p. 257: 1.7, 1.9 (a,d), 1.10 (a,c), 1.21 (a);
6) Four.I.2, p. 321: 1.1, 1.9, 1.10, 1.13, 1.15;
7) Four.III.1, p. 357: 1.14, 1.16 (a,b);
8) Five.II.2, p. 396: 2.6, 2.7 (a), 2.8, 2.11;
9) Five.II.3, p. 405: 3.22 (a,d,e), 3.23 (a,b), 3.24, 3.26(a), 3.30, 3.32;
10) Three.VI.2, p. 275: 2.10(a,b), 2.11, 2.12, 2.15; (in 2.1112 get orthonormal basis through GrammSchmidt).



1213

Quiz10 on Thursday, May 5th (Solution published). Topic: Eigenvalues and eigenvectors (see problem 6 in Exam2).
Additional reading (just for fun). MATLAB for Linear Algebra.
Three.VI.3, p. 283: 3.11(a,f);
Five.II.1, p. 391: 1.7, 1.8;
Five.II.2, p. 396: 2.6, 2.7 (a), 2.8, 2.11;
Five.II.3, p. 405: 3.22 (a,d,e), 3.23 (a,b), 3.24, 3.26(a), 3.30, 3.32.

May 6th, 2022, 5pm end of day.

1011

Quiz08 on Tuesday, May 3rd (Solution published). Topic: GramSchmidt ortogonalization process.
Quiz09 on Tuesday, May 3rd (Solution published). Topic: Least Squares method (Extra info for least squares, useful for the quiz.).
Addition to the homework. Complex numbers.
Four.III.1, p. 357: 1.14, 1.16 (a,b);
Four.Cramer's Rule, p. 360: 1(a), 2;
Three.VI.1, p. 270: 1.6(a), 1.7(a);
Three.VI.2, p. 275: 2.10(a,b), 2.11, 2.12, 2.15; (in 2.1112 get orthonormal basis through GrammSchmidt)
Three.VI.3, p. 283: 3.11(a,f), 3.14, 3.15;
Three.Line of the Best Feet  read the topic, read the Extra info for the quiz above and do the example there by yourself.

May 2nd, 2022, end of day.



Exam 2 on Thursday, April 21st (Solution published). Topics: all covered material since Exam 1, up to and including Four.III.1.
As usual, no calculators or books allowed. You can prepare a page of reference of any formulae etc. you like.
Training set of problems for Exam 2:
Three.II.2, p. 200: 2.23 (a), 2.27;
Three.III.1, p. 211: 1.18, 1.20 (a,b), 1.21(b);
Three.III.2, p. 220: 2.14, 2.19;
Also review Quiz 05;
Three.IV.4, p. 252: 4.15 (a,c,d), 4.16;
Three.V.1, p. 257: 1.7, 1.9 (a,d), 1.10 (a,c), 1.21 (a);
Three.V.2, p. 265: 2.13, 2.16, 2.28 (a);Have a look at the solution of the quiz 06!
Four.I.2, p. 321: 1.1, 1.9, 1.10, 1.13, 1.15;
Four.I.3, p. 335: 3.38;
Four.III.1, p. 357: 1.14, 1.16 (a,b).
Also review the nullspace and how to find a basis for the null space for a transformation given by the matrix.
Here is an example, study it thoroughly!

None.

9

Quiz06 on Tuesday, April 19th (Solution published). Topic: you will need to be able to change matrix map representation for different bases.
Quiz07 on Tuesday, April 19th (Solution published). Topic: determinants. (Yes, double quiz.)
Three.V.1, p. 257: 1.7, 1.8 (a,c), 1.9 (a,d), 1.10 (a,c), 1.16, 1.19, 1.21 (a);
Three.V.2, p. 265: 2.10 (a,b), 2.11 (a,b), 2.13, 2.15 (a,b), 2.16, 2.28 (a);
Four.I.1, p. 321: 1.1, 1.3, 1.4, 1.5 (b), 1.7, 1.9, 1.10, 1.13, 1.15;
Four.I.2, p. 326 (read examples in this section): 2.8(a), 2.9, 2.11, 2.20, 2.22;
Four.I.3, p. 335: 3.17, 3.19, 3.22, 3.23, 3.29, 3.38;
Four.I.4, p. 345: 4.11, 4.13;
Four.II.1, p. 350: 1.9(a,c), 1.14 (a,c), 1.19;
Four.III.1, p. 357: 1.11, 1.12, 1.14, 1.16 (a,b), 1.21, 1.27.

April 19 20th, 2022, class time.

78

Quiz02 on Tuesday, April 5th (Solution published). Topic: GaussJordan reduction method for systems of linear equations.
Quiz05 on Thursday, April 7th (Solution published). Topic: you will need to prove that some map is homomorphism by applying the definition and find a matrix representation of the map.
Three.II.2, p. 200: 2.21 (a,d), 2.22 (a,b), 2.23 (a), 2.27, 2.31 (a);
Three.III.1, p. 211: 1.12 (a,b,c), 1.13 (b), 1.15, 1.16, 1.17, 1.18, 1.20 (a,b), 1.21(b), 1.26 (ad);
Three.III.2, p. 220: (read corollary 2.6) 2.12 (c), 2.13 (b), 2.14, 2.15 (a), 2.16, 2.19;
Three.IV.1, p. 226: 1.8 (a,e), 1.15, 1.16, 1.17;
Three.IV.2, p. 233: 2.14 (b,c), 2.16, 2.18, 2.24, 2.25, 2.27 (Hint: for the last two problems use induction);
Three.IV.3, p. 243: 3.25 (a,c,e), 3.27, 3.40, 3.44, 3.45;
Three.IV.4, p. 252: 4.1214, 4.15 (a,c,d), 4.16, 4.17, 4.18, 4.20, 4.21, 4.29, 4.33 (c,d).

April 7 12 13th, 2022, end of day.



Exam 1 on Tuesday, March 8th (Solution published). Topics: all covered material.
As usual, no calculators or books allowed. You can prepare a page (one side) of reference of any formulae you like (no problems solution!).
You will need to submit this page with your exam.
Training set of problems for Exam 1:
One.I.3, p. 33: 3.15 (a,c,e), 3.16 (a,b,d);
One.II.1, p. 41: 1.2, 1.3;
One.II.2, p. 47: 2.20, 2.38, go through proofs in this section;
One.III.1, p. 54: 1.12 (a,b,d);
One.III.2, p. 62: 2.27;
Two.I.1, p. 86: 1.19, 1.22 (a,c,d);
Two.I.2, p. 97: 2.22, 2.23;
Two.II, p. 109: 1.20 (a,c), 1.21 (b,d);
Two.III.1, p. 118: 1.18 (a,b), 1.19 (a,b), 1.20 (a,b);
Two.III.2, p. 125: 2.17;
Two.III.3, p. 133: 3.18 (b), 3.19 (a,c), 3.21 (a,c,d), 3.23(b,c), 3.26;
Three.I.1, p. 172: 1.12 (b), 1.15, 1.16, 1.17 (a,c,d);
Three.I.2, p. 181: 2.12 (a,c,d), 2.18.



6

Three.I.1, p. 172: 1.12 (b), 1.15, 1.16, 1.17 (a,c,d), 1.18, 1.30, 1.37;
Three.I.2, p. 181: 2.9 (a,b,d), 2.12 (a,c,d), 2.18;
Three.II.1, p. 188: 1.18 (a,c,d), 1.19 (b,c), 1.20, 1.22, 1.23, 1.29, 1.30.

March 3rd 8th 11th, 2022, end of day.

5

Quiz03 on Tuesday, March 1st (Solution published). Topic: Linear dependence and independence.
Quiz04 on Tuesday, March 3rd (Solution published). Topic: Vector Spaces and Linear Systems (review Tuesday's lecture on columns relations, etc.).
Two.III.1, p. 118: 1.18 (a,b), 1.19 (a,b), 1.20 (a,b), 1.21, 1.23, 1.25 (a,c), 1.28 (a,b), 1.34;
Two.III.2, p. 125: 2.16, 2.17, 2.18 (a,c), 2.21, 2.22, 2.23, 2.26;
Two.III.3, p. 133: 3.16 (a,c,e), 3.17 (b), 3.18 (b), 3.19 (a,c), 3.20, 3.21 (a,c,d), 3.23, 3.26, 3.37.

February 25th March 2nd, 2022, end of day.

4

One.III.1, p. 54: 1.8 (a,d), 1.10 (a,b,c), 1.12 (a,b,d), 1.15;
One.III.2, p. 62: 2.11 (a,b,c,e), 2.21, 2.24;
Two.I.1, p. 86: 1.17, 1.18, 1.19, 1.22 (a,c,d), 1.24, 1.25, 1.28 (b,c,e), 1.35(b), 1.38;
Two.I.2 (read examples in this section), p. 97: 2.20 (a,b), 2.21, 2.22, 2.23 (b), 2.24 (a,b,d), 2.25 (a,c,e), 2.27 (a,c,e), 2.28, 2.34;
Two.II, p. 109: 1.20 (a,c), 1.21 (b,d), 1.22 (a,c), 1.23 (a,b), 1.24 (a,b,d), 1.27, 1.28 (a), 1.35, 1.42 (a).

February 18th, 2022, class time end of day.

23

One.I.3, p. 33: 3.15 (a,c,e), 3.16 (a,b,d), 3.18 (a), 3.19, 3.20 (a,c,e), 3.21 (a,d), 3.22;
One.II.1, p. 41: 1.2, 1.3;
One.II.2, p. 47: 2.11 (a,c,e), 2.12, 2.13, 2.14, 2.15, 2.20, 2.25, 2.38.

February 1st 10th, 2022, class time.

1

Quiz01 on Thursday, January 27th February 1st (Solution published). Topic: GaussJordan reduction method for systems of linear equations.
One.I.1, p. 9: 1.17 (a,b), 1.18 (af), 1.19, 1.20, 1.22, 1.25, 1.28, 1.33;
One.I.2, p. 20: 2.15, 2.16, 2.18 (a,d,e), 2.19 (a,c), 2.20 (a,b), 2.23, 2.24, 2.27(a,b).

January 25th, 2022, class time.

