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HW set Lecture From Due Problems
I 1 05/01 14/01 1.2  9, 18
2,3 09/01 14/01 1.3  11, 12, 14, 16
(everywhere omit F)
II 4 12/01 21/01 1.4  2(c), 3(a,c), 5(f)
5 14/01 21/01 1.4  12   1.5  2(c), 9  Bonus: 1.5  18
6 16/01 21/01 1.6   3(a,c), 4, 7
III 7 20/01 28/01 1.6  8, 9, 13, 16
8,9 24/01 28/01 2.1  8, 9, 13, 19, 40(a,b)
Bonus: 2.1  22 (only show the existence of scalors a,b,c.)
IV 10 27/01 04/02 2.2   2(d), 3, 4, 5(b)
11 28/01 04/02 2.3  2(a), 3(a), 12
12 30/01 04/02 2.4  2(a,b,c,d), 3(a), 4, 6
V 13 02/02 11/02 2.5  2(b,d), 3(a,e)  
14  04/02  11/02 2.5   4, 5   3.1  2, 7
15  06/02 11/02 3.2  2(a,d,g), 4, 11 
VI 16  10/02 18/02 3.2 3, 5(b,e), 6(b), 8 
17   18/02 no problems
18   18/02 no problems
VII 19  18/02 25/02 3.4 2(h), 10  4.2 12, 23, 27
20  18/02 25/02 4.3  7, 12, 21  5.1  3(a,d), 4(c), 20
21 20/02  25/02 5.1  7(a), 11(c),  22 
VIII 22 26/02  4/03 5.2  2(f,g), 3(a,d), 12  
23 26/02  4/03 6.1  3, 6, 10, 12 
24  27/02 4/03 6.2  2(a,c,d), 7, 8,15(a)
IX 25 03/03  9/03 6.3  2(a,c), 3(a,c), 4, 12
26 03/03  9/03 6.4  2(a,d), 5, 6
27  06/03 9/03 6.5  3,7



Remark.  Make sure that each Wednesday you submit homeworks assigned for each of the three lectures of the previous week. 

Remark.  The plan for future lectures is tentative. It will change irregularly.  

Problem 2.2 40(a,b)   (see the comment in the textbook before Problem 40.)
Here  v+W  is defined to be
{ v+w | w is in W }.
In other words, v+W is a subset of V (it is not a subspace unless v=0) that consists of all vectors of the form v+w where w is in W.


Remark:  In all problems the field F should be replaced by real numbers R.

In general we may work not only with real numbers, but with more general numbers ("fields"), say, complex numbers. For example, we may consider matrices whose entries are complex numbers, we may consider functions with complex values, or we may consider polynomials with complex coefficients. If you are interested in more details, you may want to read Appendix C in the textbook. On the other hand in Lecture Notes I will continue (for a while) to use only real numbers, so in all homework problems the field F should be replaced by real numbers R. 


Problem 1.5(2c)  
Recall that  P3 is the space of polynomials of degree less than or equal to 3. 


Problem 1.2(9)
(a)  In Lecture Notes in the definition of a vector space replace the axioms

(A3)   There is a unique vector 0 such that A + 0 = A for each vector A
(A4)    For each vector A, there is a unique vector -A such that A + (-A) = 0

by new axioms

(A3')   There is a vector 0 such that A + 0 = A for each vector A
(A4')    For each vector A, there is a vector -A such that A + (-A) = 0.

Show that the new definition of a vector space is equivalent to the original definition. In other words show that the definitions of a vector space in the textbook and in the Lecture Notes are equivalent.

(b) Show that  a0=0 for each real number a.

Problem 1.2(18)  
Let  V={(a1 , a2) |  a1, a2 are real numbers}. For (a1 , a2), (b1 , b2) in V and a number c define
(a1 , a2) + (b1 , b2) = (a1 + 2b1 , a2 + 3b2)
and
c(a1 , a2)=(ca1 , ca2).       
Is V a vector space?  Justify your answer.