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 not there, slide your exam under the door. Your exam m
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 P2 (R):
1
1
p, q :=
2
p(x)q (x)dx
1
(a) (10 points) U
MATH 113: PRACTICE FINAL
Note: The nal is in Room T175 of Herrin Hall at 7pm on Wednesday, December
12th.
There are 9 problems; attempt all of them. Problem 9 (i) is a regular problem, but 9(ii)-(iii)
are bonus problems, and they are not part of your regu
MATH 220: PDES AND BOUNDARIES
We have used the Fourier transform and other tools (factoring the PDE) to solve
PDEs on Rn . We now study how we can use these results to solve problems on the
half space, or indeed on intervals, cubes, etc.
As you have shown
MATH 220: SOLVING PDES
We now return to solving PDE using duality arguments and energy estimates.
Before getting into details, we note that the ideal kind of well-posedness result
we would like is the following. We are given a PDE (including various addit
MATH 220: THE FOURIER TRANSFORM TEMPERED
DISTRIBUTIONS
Beforehand we constructed distributions by taking the set Cc (Rn ) as the set
of very nice functions, and dened distributions as continuous linear maps u :
Cc (Rn ) C (or into reals). While this was a
Math 113 Homework 8
Graham White
May 31, 2013
Book problems
1. From the previous homework set, we know that an orthonormal basis for P2 (R) with the given inner product is
p
p
cfw_1, 3(2x 1), 5(6x2 6x + 1). The matrix of T with respect to this basis is
p
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
ak T k v = ak+1 T k+1 v + . . . + am
1T
1
v is linearly de
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 , 1i1
hx2 ,
p
hx, 1i1 = x
1
2.
1
)i(x
2
1
) = x2
2
12(x
Nor
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 1, because U 6= cfw_0. Extend this to a basis of V ,
c
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.
T (v)
T (u) u)
Let v be any element of V . Then v = (v
MATH 113: PRACTICE FINAL SOLUTIONS
Note: The nal is in Room T175 of Herrin Hall at 7pm on Wednesday,
December 12th.
There are 9 problems; attempt all of them. Problem 9 (i) is a regular problem, but
9(ii)-(iii) are bonus problems, and they are not part of
MATH 113: PROBLEM SET 8
Do Axler, p.122-124:3,5,6,7,9,10,17,20,22, and the following two problems.
Problem 1. In this problem we study direct sums of vector spaces without having an ambient
vector space in which they lie.
Let U, W be vector spaces over F,
MATH 113: PROBLEM SET 9
Do Axler, p.125:24,26,27,28,30, p.158:4,11,14,17, p.245:10,16 (for problems involving traces,
you may assume that V is complex), and the following problem.
Problem 1. Let V be a vector space over F. Recall that V = L(V, F) is the d
MATH 113: MIDTERM
Each problem is 20 points. Attempt all problems. Problem 5, part (5) is extra credit only,
so work on it only if you have completed all other problems.
This is a closed book, closed notes exam, with no calculators allowed (they shouldnt
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 v1 + a2 v2 + . . . + an v n .
Rearranging this equatio
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 + ( v) = 0.
Therefore,
(equality is transitive)
( v) +
Math 113 Homework 4
Due Friday, May 3, 2013 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your
solutions. Solutions are due to the Course Assistant, Graham White, in his office, 380-380R
(either hand your solutions
Math 113 Solutions: Homework 7
November 14, 2007
1.) Suppose T is invertible with inverse S. Then ST = T S = I so, for any particular
basis (v),
Mat (T, (v) Mat (S, (v) = Mat (T S, (v) = Mat (I, (v) = I
and similarly,
Mat (S, (v) Mat (T, (v) = Mat (ST, (v
Math 113 Homework 1
Due Friday, April 12th, 2013 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your
solutions. Solutions are due to the Course Assistant, Graham White, in his office, 380-380R
(either hand your solut
Math 113 Homework 3
Due Friday, April 26th, 2013 by 4 pm
Please remember to write down your name and Stanford ID number, and to staple your
solutions. Solutions are due to the Course Assistant, Graham White, in his office, 380-380R
(either hand your solut
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 the set of pairs of elements
(u, v), where u, v V . A
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 space. Let U1 , . . . , Um be subspaces
of V whose sum is d
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 multiplication by
(af )(x) = a(f (x) for f, g FX and a
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, prove that
if V = Null(T ) + Null(T I), then T 2 = T