MAT 310
Linear Algebra
Midterm 1 Solutions
1. Write a complete, clear and correct proof of the following statement: If
a list (v1 , v2 , . . . , vn ) of elements of a vector space is linearly dependen
REVIEW FOR MIDTERM: MAT 310, SPRING 2014
The midterm will take place the Thursday lecture on 3/13, in our usual
lecture room (P-118). The exam will cover all sections 1.1-1.7 and 2.1-2.5
of our text b
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LINEAR ALGEBRA MAT 310: SPRING 2014
Topics and Text:
This course will focus on mathematical proofs as well as on computations.
Some of the topics we will cover are: matrics and operations on matrices;
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Homework 5 Solutions
Math 131A-3
1. Problems from Ross.
(17.3) (a) By assumption, cos(x) is continuous, so since products of continuous functions are continuous, cos4 x is continuous. Moreover, consta
REVIEW FOR THE FINAL EXAM: MAT 310, SPRING 2014
Here are 9 review problems listed below of a type similar to those which
will be on the nal exam.
0
(1) Set A = 0 2 0 . For which values of is A diago
21. sqrt(n) < (1/sqrt(1)+ (1/sqrt(2)+. (1/sqrt(n)
37. On the outside rim of a circular disk the integers from 1 through 30 are painted in random order.
Show that no matter what this order is, there mu
MAT 310
HW 5
Section 2.3
Exercise 2
20 9 18
a) A(2B + 3C) =
5 10 8
29
A(BD) = (AB)D =
26
2
3
4
b) At =
5 1 2
23 19 0
At B =
26 1 10
12
t
BC = 16
29
CB = 27 7 9
CA = 20 26
Exercise 12
(a) Let v V s
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UCB Math 110, Fall 2010: Homework 9
Solutions to Graded Problems
5.3.4 We can write A = Q1 DQ, with some invertible Q and diagonal D. Since the limit
L = lim Am = Q1
m
lim Dm Q
m
exists, each diagonal
REVIEW FOR THE FINAL EXAM: MAT 310, SPRING 2014
Here are 9 review problems listed below of a type similar to those which
will be on the nal exam.
0
(1) Set A = 0 2 0 . For which values of is A diago
HW 6:MAT 310, SPRING 2014
(A) Let denote a subset of the vector space V. We have discussed in class
what it means for to be a maximal independent subset of V (see section
1.7). Give a denition for wha
FIELD OF FRACTIONS
The eld F is contained in a natural way in the set of polynomials over
F, P (F ), as the constant polynomials. The set of polynomials P (F ) is also
contained in a eld E (called the
FINAL EXAM: MAT 310, SPRING 2014
Instructions: Complete each of the following 6 problems in the spaces
provided. Please either show your work or give an explanation for each
of your answers: this is a
HW 10:MAT 310, SPRING 2014
Do problems #10,12,14,15 in section 5.1, in addition to problems A,B,C,D
below. At least two of the problems A,B,C,D below will be among those
graded for this assignment.
(A
HW 11:MAT 310, SPRING 2014
Complete the following four problems. In these problems T : V V
denotes a linear transformation with V an n-dimensionsal vector space over
the eld F.
(1) Let T : V V denote
HW 12:MAT 310, SPRING 2014
Complete the following 3 problems.
(1) A polynomial p(x) P (F ) with coecients in the eld F is irreducible
if whenever p(x) = r(x)s(x) for r(x), s(x) P (F ) we have that eit
END OF SEMESTER NOTES FOR APPENDIX E AND
SECTION 7.4: MAT 310, SPRING 2014
Here are a few ideas about polynomials which were covered in class but
were not completely contained in Appendix E.
Deninitio
LINEAR ALGEBRA MAT 310: SPRING 2014
Topics and Text:
This course will focus on mathematical proofs as well as on computations.
Some of the topics we will cover are: matrics and operations on matrices;
1
1
0
0
(1) Let denote the set of vectors v1 = , v2 =
1
1
0
1
1
1
in 4-space R4 .
0
0
, v3 =
(a) (10 points) Explain why is a basis for a subspace V R4 .
Solution: If for real numbers
MAT 310
HW 3
Section 1.5
Exercise 1
a) False, possibly just one.
b) True, since cfw_0 is already linearly dependent.
c) False, by definition.
d) False. For example, cfw_e1 , e2 cfw_e1 , e2 , e1 + e2
If the Woombia ' n is zero, bhhehicientity princip for polyno ials we must have in
particular at an = 0, a ontradiction. herefore S is lin arly indepe ent.
Section 1.6
Exercise 1
a) False, cfw_O is a
Math 310 MIDTERM II FALL 2016
LAST NAME, NAME, ID:
RECITATION: (circle one) : 01 (Minoccheri) 02 (Yuan)
Problem Points Possible Points
Total 200
INSTRUCTIONS:
1.
Show your work.
Math 310 MIDTERM I FALL 2016
1. (30 pts) A real function f dened on the real line R is said to be an odd function if
t) : f(t), Vt E R. Prove that the set 0 of odd functions with the operations of
add