Math 113 Winter 2013 Prof. Church
Final Exam: due Monday, March 18 at 3:15pm
Name:
Student ID:
Signature:
Your exam should be turned in to me in my oce, 383-Y (third oor of the math building). If I
am
Math 113 Final Exam: Solutions
Thursday, June 11, 2013, 3.30 - 6.30pm.
1. (25 points total) Let P2 (R) denote the real vector space of polynomials of degree
2. Consider the following inner product on
Math 113 Homework 5
Graham White
May 11, 2013
Book problems:
3. For the sake of a contradiction, assume that the set cfw_v, T v, T 2 v, . . . , T m
some integer k with 0 k m 1 there is an equation
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EXERCISES
the given vector space axioms hold in the respective given sets:
+(+v) = (+I3)+Y foraav eY
m)cc=k(moc) aeA, k,me R '
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c.) S = cfw_(xy) I x,y e R together with the operation
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Math 113 Solutions: Homework 9
December 6, 2007
p. 125
24.) Let T : P2 (R) R be dened by T (p) = p(1/2). This is a linear functional on
P2 since addition and scalar multiplication on P2 are dened poin
Math 113 Solutions: Homework 8
November 28, 2007
c1
d1
c2
d2
1
3.) Dene cj = jaj , dj = j bj , for j = 1, 2, ., n. Let c = . and d = .
.
.
.
.
cn
dn
vectors in Rn . Then the Cauchy-Schwarz inequ
Math 113 Solutions: Homework 5
October 31, 2007
9.) Let 1 , 2 , ., be an enumeration of the non-zero eigenvalues of T . For each i
1
nd eigenvector vi = 0 so that T vi = i vi . Now vi = T i vi range (
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Math 113 Homework 7
Graham White
May 28, 2013
Book problems
10. The rst element of our basis is the function 1.
The second element is proportional to x
The third element is proportional to
x2
hx2 , 1i
Math 113 Homework 4
Graham White
May 5, 2013
Book problems:
3. The claim is true. Let U be any subspace of V other than cfw_0 and V itself. Then U has a basis cfw_u1 , u2 , . . . , um .
We have that m
Math 113 Homework 3
Graham White
April 30, 2013
Book problems:
4. To show that V = null(T ) cfw_au, a 2 F, we need to show that V = null(T ) + cfw_au, a 2 F and that
null(T ) \ cfw_au, a 2 F = cfw_0.
Math 113 Homework 2
Graham White
April 20, 2013
Book problems:
1. Consider an arbitrary element v of V . The set cfw_v1 , v2 , . . . vn spans V , so there are scalars a1 , a2 , . . . , an with
v = a1
Math 113 Homework 1
Graham White
April 20, 2013
Book problems:
3. Let v be an arbitrary element of the vector space V . By the denition of the additive inverse, we have that
( v) + ( v) = 0
and that
v
Axler, Chapter 3: 4, 11, 14, 23, 24, and the following problems.
6. Let T L(V ), and write T 2 = T T , the composition of T with itself.
Prove that if T 2 = T , then V = Null(T ) Null(T I). Conversely
Homework due 1/12: 1.3, 1.4, 1.11, 1.12, 1.14, and the following exercises.
1. Let X be a set and write FX for the set of functions X F. Define addition by (f + g)(x) = f (x) + g(x) and define scalar
Axler, Chapter 2: 1, 2, 5 (first read the definition of F in the book), 11,
12 (first read the definition of Pm (F ) in the book), 17, and these exercises:
7. Let V be a finite dimensional vector spac
Axler, Chapter 3: 17, 18, 20 (I stated this in class without proof), 21.
Chapter 5: 1, 2 (do not assume that the collection is finite), 8, and this exercise:
8. Let V be a vector space. Let V V denote
Math 113 Syllabus
Fall 2013
Lecture: Tuesday, Thursday 2:15-3:30pm 380-380F Sloan Hall Basement
Course Webpage: http:/math.stanford.edu/~dankane/113/
Professor: Daniel Kane
Email: dankane at math.stan
CHAPTER 1: VECTOR SPACES AND SUBSPACES.
1.1. VECTOR SPACES.
Definition:
A field is a set
(i) For all
together with two operations
the sum
and for which the following axioms hold:
and the product
(ii)F
Math 113 Homework 4 solutions
May 2017
Page 139, Problem 2
Suppose S, T L(V ) are such that ST = T S. Prove that null(S) is invariant
under T .
Solution. Assume v null(S). This means Sv = 0. But then
Math 113 Homework solutions
April 2017
Question 10
Suppose U1 and U2 are subspaces of V . Prove that the intersection U1 U2 is
a subspaces of V .
Solution.
The origin 0 U1 and 0 U2 , so 0 U1 U2
Supp
Midterm 1, Math113.
The test is close book, close note, and no calculator. The only information
allowed is a one-sided sheet of information prepared. Please show reasonable work
to support your answer
Midterm 1, Math113 Solution.
Problems
1. Let T : V W be an invertible linear map, ane let v1 , , vn V be vectors
in V . Show that when T v1 , , T vn is a basis of W , then necessarily v1 , , vn
is a b