PHY152 Lecture
March 16, 2016
If we consider the electromagnetic wave (light)
2f
1 2f
=
,
x2
c2 t2
we can write the wave function as f (x, t) = f (x ct).
Suppose u = x ct, then
f
u f
=
.
x
x u
Yet, u/x = 1, so
f
f
=
.
x
u
And
2f
2f
=
.
2
x
u2
on the right
MAT247
Denitions (part 2)
Winter 2014
A basis of a vector space V is a collection of vectors cfw_vi iI such that for all v V , there exists a unique
collection of scalars cfw_ai iI such that v = iI ai vi and at most, nitely many ai are non-zero.
A subset
Multilinear Forms
A multilinear form on a vector space V(F) over a field F is a map
f:V(F).V(F)->F
(1)
such that
cf(u_1,.,u_i,.,u_n)=f(u_1,.,cu_i,.,u_n)
(2)
and
f(u_1,.,u_i,.,u_n)+f(u_1,.,u_i^',.,u_n)
=f(u_1,.,u_i+u_i^',.,u_n)
(3)
for every c in F and any
Quad Form
A quadratic form involving n real variables x_1, x_2, ., x_n associated with the nn
matrix A=a_(ij) is given by
Q(x_1,x_2,.,x_n)=a_(ij)x_ix_j,
(1)
where Einstein summation has been used. Letting x be a vector made up of x_1, ., x_n
and x^(T) the
Tensor
An nth-rank tensor in m-dimensional space is a mathematical object that has n indices
and m^n components and obeys certain transformation rules. Each index of a tensor
ranges over the number of dimensions of space. However, the dimension of the spa
Symmetric Forms
A symplectic form on a smooth manifold M is a smooth closed 2-form omega on M
which is nondegenerate such that at every point m, the alternating bilinear form
omega_m on the tangent space T_mM is nondegenerate.
A symplectic form on a vecto
MAT 247 midterm
Name:
February 15, 2011
1. Let V, , be an inner product space. Let W V be a subspace.
(a) Give the denition of W , the orthogonal complement of W .
(b) Suppose that W = V . Prove that W = cfw_0.
1
2. Consider R3 with the usual inner produc
MAT247HS
MIDTERM
March 5, 2013
Please write clearly and show all of your work.
No notes or calculators are allowed.
Each problem is worth 25 points.
1. Let W be the subspace of R4 spanned by the vectors
w1 = t (1, 1, 0, 1), w2 = t (2, 1, 2, 0).
(i) Find a
Midterm Solutions
1. (i) We nd an orthogonal basis cfw_v1 , v2 , v3 for W , to avoid having to carry around some
square roots. So
v1 = w1 = t (1, 1, 0, 1, 0),
v 2 = w2
w2 , v1
v1 = t (1, 0, 2, 1, 1)
v1 , v1
v 3 = w3
w3 , v1
w3 , v2
v1
v2 = t (0, 1, 2,
MAT247H1S
2015
Algebra II
Midterm Exam Information
The Midterm Exam will be held on Tuesday, March 3th, 1:103 pm in our regular classroom.
The exam will cover the material from the following section, as covered in class:
2.2, 2.4, 2.5, 5.1, 5.2, 5.4 6.1,
MAT 247H1S - ALGEBRA II
Term Test - Solutions to questions 4 and 5
4. Let W be a nite-dimensional subspace of an inner product space V . Let T L(V ) be
orthogonal projection on the subspace W .
x, x , x V .)
a) Prove that T (x) x for all x V .
(Recall tha
A symmetric bilinear form on a vector space V is a bilinear function
Q:VV->R
(1)
which satisfies Q(v,w)=Q(w,v).
For example, if A is a nn symmetric matrix, then
Q(v,w)=v^(T)Aw=<v,Aw>
(2)
is a symmetric bilinear form. Consider
A=[1 2; 2 -3],
(3)
then
Q(a_1
Inner Products
An inner product is a generalization of the dot product. In a vector space, it is a way to
multiply vectors together, with the result of this multiplication being a scalar.
More precisely, for a real vector space, an inner product <,> satis
PHY152 Lecture
March 14, 2016
1D Waves
The continuum/field version of SHM
Fundamental model of physics
Type of Waves
mechanical wave
electromagnetic wave
matter wave (quantum mechanics concept)
gravitational wave
longitudinal vs. transverse wave
l
MAT157 Lecture
March 17, 2016
Uniform Convergence and Power Series
limits of sequences of functions:
Example:
x n
)
n
Relation between functions and infinite sums:
ex = lim (1 +
n
Example:
f (x) =
n
X
f (k) (a)
k=0
k!
(x a)k + Ran f (x)
lim Ran f (x) f (x
Convergence of Series
March 15, 2016
P
an is absolutely convergent if n=1 |an | converges.
P
theoremvery absolute convergent series is convergent. Series
aP
n is absolutely convergent if and only
if
series
formed
from
positive
terms
a+
n and
P
formed fro
MAT247 Lecture
March 24, 2016
Let V be a finite-dimensional vector space over F . And T : V V is
a linear operator such that PT (t) = (1 t)m1 .(k t)mk splits completely,
1 , ., k are distinct eigenvalues and m1 , ., mk are the multiplicities for the
corre
MAT247 Lecture
March 22, 2016
Jordan Canonical Form (JCF)
Definition 1. A Jordan block ( block) is a r r matrix B of the form
1 0
0 0
. . . .
.
. .
. .
0 0
(1)
for some .
A matrix A is in JCF if
.
.
B1
.
A= .
O
O
.
.
(2)
Bk
each Bj is a Jordan bloc
MAT247 Lecture
March 17, 2016
Let V be a finite-dimensional vector space over F , dimV = n and T is a
linear operator, there is a v V and v 6= 0, the T -cyclic subspace generated by
v W = span(cfw_v, T (v), T 2 (v), .).
Suppose k = dimW , where 1 k n, the
MAT247 Lecture
March 15, 2016
Theorem 1. (Spectral Theorem) Let V be a finite-dimensional complex inner
product space. T is a normal operator with 1 , ., k distinct eigenvalues. Then
W1 , ., Wk are the eigenspaces of T for which Wj is the eigenspace corre
Midterm review
MAT 247
The midterm will be held on Tuesday March 4, in EX 300 (thats the
exam centre), 1:15 pm - 3 pm.
The midterm will cover all material presented in class before reading
week, as well as Assignments 1-6.
Study the following denitions: g
Math 247S
Practice Term Test
Winter 2012
Rules: No books, no notes. You have 50 minutes to complete the test. Note: the actual test is
shorter than the practice test.
(1) Let V be a complex vector space with two inner products , 1 and , 2 .
Suppose v, v 1
Practice Final
(1) Recall that for a group G its center Z(G) is dened to be the set of all elements
h G such that hg = gh for any g G.
(a) Prove that Z(G) is a normal subgroup of G.
(b) Show that Z(Sn ) is trivial for any n > 2.
(c) Find the center of GL(
MAT247S, 2009 Winter, Problem Set 5 Solution
Grader: TAM, Kam Fai, [email protected]
1. By nite dimensionality, one can write W = n F (T n (x) for some
i=0
nite n. Then (a) and (b) are straightforward.
2. As hinted, write V = n F (T n (x) and U (x) = g(
MAT247S, 2009 Winter, Problem Set 4 Solution
Grader: TAM, Kam Fai, [email protected]
4 0 0
1 1 1
1 0 2 , D = 0 2 0 .
0 0 2
1 1 1
Ti1 ,.,ik = orthogonal projection
1.(a) P =
on the space spanned by cfw_vi1 , . . . , vik ,
(b) Let
where vj is the j-column
MAT247S, 2009 Winter, Problem Set 3 Solution
Grader: TAM, Kam Fai, [email protected]
1.(a) Just check AA = A A. (b) Let = exp(i/4) be the 8-th roots
of unity. A possible orthonormal basis is cfw_
with eigenvalues , , , respectively.
0
1
0
,
0
0
1
,
0
1
MAT301 Groups and Symmetries
Practice Problems 3
1
MAT301 Practice Problems Part 3
1. In each case below, prove or disprove that the subgroup H is normal in the group G. If
H is normal in G, determine whether the factor group G/H is abelian. (Note: You do
MAT 301F - Problem Set 2
Due Wednesday October 21, 2015 at 6:10pm.
Please explain your answers. Correct answers with incomplete explanations will not receive
full marks.
1. Assume that n 3. Let H = cfw_ Sn | (3) (1) . Prove or disprove that H is a
subgrou
MAT 301 Practice problems part 3
1. In each case below, prove or disprove that the subgroup H is normal in the group G.
If H is normal in G, determine whether the factor group G/H is abelian. (Note: You
do not need to show that H is a subgroup of G.)
a
b
MAT301 - Groups and Symmetries
University of Toronto, Prof. F. Murnaghan
Hints and solutions for problem set #3
TA: Jerrod M Smith
November 30, 2015
Hints and solutions
1. In each case below, prove or disprove that the subgroup H is normal in the group G.