Physics 115A Problem Set #3 Solutions
Sean Litsey
April 17, 2015
Solutions to Griffiths problems are based on the official solutions.
Problem 1 (Griffiths 1.3)
Part a
The integral can be performed by a u-substituion of u = x a:
2
1 =
Ae(xa) dx
2
=
Aeu
Homework #5
Read Homan and Kunz Sections 3.5 and 3.6. We shall go over material from these
sections in the 4:00 session Wednesday.
Problem 1. For each of the following linear transformations T , compute the rank of
T := dim (im T ), the nullity of T := di
Direct Sums of Subspaces and Fundamental
Subspaces
S. F. Ellermeyer
July 21, 2008
1
Direct Sums
Suppose that V is a vector space and that H and K are subspaces of V such
that H \ K = f0g. The direct sum of H and K is the set of vectors
H
K = fu + v j u 2
Take Home Midterm and Homework #7
Instructions: This homework/midterm is open book. All proofs must be fully written or
typed out. You may not cut and paste solutions that you did not write or type yourself.
If you copy part of or the whole proof from a r
1. Let F be a field and let f be the linear functional on F 2 defined
by f (x1 , x2 ) = ax1 + bx2 . For each of the following linear operators T , let
g = T t f , and find g(x1 , x2 ).
( Hint: Simply you can apply Theorem 21 on page 112 to solve this
ques
Math 115AH, Final, June 9, 2009
1. a) Let x1 , x2 , . . . . , xn be a set of vectors in V . Define what it means
to say x1 , x2 , . . . , xn are linearly independent.
Comments: A basic definition
b) Suppose x, y, z 2 V . Prove that x, y, z are are linearl
Math 115AH, Final, Dec. 10, 2010
1. If S is a subset of V , we define S 0 is the set of all elements of V which
vanish on S.
a) If V = R3 and
Find a basis of S 0 .
80 1 0
>
>
1
1
>
<B C B
B C B
S = B 2 C, B 0
>
@ A @
>
>
: 3
1
19
>
>
C>
=
C
C .
A>
>
>
;
C
Math 115AH, Final, Dec. 5, 2011
1. Suppose B = cfw_1 , 2 , . . . , n is a basis of V and T : V ! V and 1
is an eigenvector of T with eigenvalue
1.
What is the first column of [T ]B ?
Comments: Easy. You can apply the definition of an eigenvalue-eigenvect
Section 6.4 Invariant subspaces
2
1. Let T be the linear
0 operator
1 on R , the matrix of which in the standard
ordered basis is A = @
1
1
A.
2 2
(a) Prove that the only subspace of R2 invariant under T are R2 and the
zero subspace.
Comments: This is sim
Math 115AH, Final, Dec. 13, 2012
1. In this problem, F = C. Suppose V has an inner product < , > and
that 1 , . . . , n are an orthonormal basis of V . Suppose W also has an inner
product < , > and that
1, . . . ,
m
are an orthonormal basis of W . Suppose
Section 6.7 Invariant direct sums
1. Let E be a projection of V and let T be a linear operator on V . Prove
that the range of E is invariant under T if and only if ET E = T E. Prove
that both the range and the null space of E are invariant under T if and
Math 115AH, Final, June 8, 2011
1. If S is a subset of V , we define S 0 is the set of all elements of V which
vanish on S.
a) If V = R3 and
Find a basis of S 0 .
80 1 0
>
>
1
1
>
<B C B
B C B
S = B 0 C, B 1
>
@ A @
>
>
: 3
1
19
>
>
C>
=
C
C .
A>
>
>
;
Co
Section 8.2 Inner Product Spaces
2. Apply the Gram-Schmidt process to the vectors 1 = (1, 0, 1), 2 =
(1, 0, 1), 3 = (0, 3, 4), to obtain an orthonormal basis for R3 with the standard inner product.
Comments: The Gram-Schmidt process is described in the pr
Section 6.2 Characteristic Values
4. Let T be the linear operator on R3 which is represented in the standard
ordered basis by the matrix
9 4 4
8 3 4
16 8 7
Prove that T is diagonalizable by exhibiting a basis for R3 , each vector of
which is a character
Homework #4
Problem 1. let V be a vector space over F . Let T : F V be a linear map. Let
v = T (1). Show that T () = v for any F .
Problem 2. Let T : V W and S : W X be linear transformations. Then S T :
V X is a linear transformation.
Problem 3. Let T :
Homework #8
R3
Problem 1. Let V = C[1, 3] with hf, gi = 1 f g. Let f (x) = x1 . Show that the constant
polynomial g nearest f is g = 21 ln 3. Compute |g f |2 for this g.
R 2
Problem 2. Let V = C[0, 2] with hf, gi = 0 f g. Let W = cfw_1, cos x, sin x. Let
SYLLABUS
PHYSICS 115A
Quantum Mechanics
Spring 2015
Instructor: H. W. Jiang, Professor of Physics
Office: Knudsen Hall 6-130E
Phone: 310-825-1796
E-mail: [email protected]
Office Hours: W, F: 2-3
Teaching Assistant: Sean Litsey ([email protected]),
M
Math 115AH
Fall 2014
Professor: David Gieseker
Office: Math Sciences 5636. Phone: 206-6321, email: dag at math ucla edu
Philosophy of 115AH: Linear algebra is a subject the permeates almost
all of mathematics, both pure and applied. The concepts of linear
Math 115AH
Review for Final.
Heres a list of the sections we will be covering in the final.
1.1, 2.1, 2.2, 2.3, 2.4, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7. In 3.2, you can skip
Theorem 5 and Example 10, 3.3, 3.4, 3.4, 3.5, 3.6, 3.7, 6.1, 6.2, 6.3 up to
but no
Math 115AH
Review for Final, December 13.
You can have a note card, 4 x 6, both sides.
Exam will cover: 2.1-5, 3.1-5, 6.2,6.4 8.1-5. In 6.4, only the material on
page 199 will be covered. Final will cumulative.
You should know the statements of the major
Math 115AH
Final
June 9, 2009
Name:
,
Please put your last name first and print clearly
Signature:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Please do not use the spaces on this page.
NO CALCULATORS!
V is a vector space over a field F in the following proble
Math 115AH
Final
December 15, 2014
Name:
,
Please put your last name first and print clearly
Signature:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Extra Credit.
Please do not use the spaces on this page.
NO CALCULATORS!
General instructions: V is a vector spa
Math 115AH
Final
December 5, 2011
Name:
,
Please put your last name first and print clearly
Signature:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Extra Credit.
Please do not use the spaces on this page.
NO CALCULATORS!
General instructions: V is a vector space ov
Math 115AH
Final
December 10, 2010
Name:
,
Please put your last name first and print clearly
Signature:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Please do not use the spaces on this page.
NO CALCULATORS!
V is a vector space over a field F in the following p
Math 115AH
Final
December 13, 2012
Name:
,
Please put your last name first and print clearly
Signature:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Extra Credit.
Please do not use the spaces on this page.
NO CALCULATORS!
General instructions: V and W are vector sp
Math 115AH
Final
December 13, 2013
Name:
,
Please put your last name first and print clearly
Signature:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Extra Credit.
Please do not use the spaces on this page.
NO CALCULATORS!
General instructions: V is a vector space o
Homework #6
Read Homan and Kunz Section 3.7 (and finish 3.5-3.6). We shall continue going
over material from these sections in the 4:00 session Wednesday.
Problem 1. Let V be a finite dimensional real vector space of odd degree. Show that
any linear opera
Math 115AH Homework #9
Problem 1. Let V be a finite dimensional inner product space over F and T : V V
linear. Show that im(T ) = ker(T ) .
Problem 2. Let V be a finite dimensional complex inner product space and T : V V
a linear operator. Show that T is
Homework #3
Problem 1. Let V be a finite dimensional vector space over F and W a subspace of V .
Show W = V if and only if dim(W ) = dim(V ).
Problem 2. Let U and W be subspaces of a vector space V over F . Suppose that dim(U )
+ dim(W ) > dim(V ). Show U