MAT 247S - Partial solutions to problem set 8
8. a) Let = cfw_ x1 , . . . , xn be a basis for V . Let T , U be linear operators on V such that A = [T ]
is a diagonal matrix with A11 = 1 and Ajj = 1, 2 j n, and B = [U ] is a diagonal
matrix with entries B
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
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
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
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(
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
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
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
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 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 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
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
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
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
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
MAT247S, 2009 Winter, Problem Set 7 Solution
Grader: TAM, Kam Fai, geo.tam@utoronto.ca
All below I denote D to be the dot diagram for eigenvalue .
1.(a) D4
,
D1
.
(b) dim(N (T 4I)j = j for j = 1, 2, 3, 4, dim(N (T 4I)j = 4 for j 4;
dim(N (T + I) = 2, dim
MAT247H1S ALGEBRA II Term Test Solutions for 1 - 3
1.(a) Take a basis of V including (1, 0, 2), then apply Gram-Schmidt. For
example, take cfw_(1, 0, 2), (1, 0, 0), (0, 1, 0), then after Gram-Schmidt we get
1
1
1
cfw_ 5 (1, 0, 2), 5 (2, 0, 1), (0, 1, 0).
MAT247 - Problem Set 2 Solution
Problem 1
a) We can easily pick {:c, 1.2} as a. basis. Using Gram-Schmidt to get a
orthonormal basis. Take inner product of T with this basis we will get
T(z:) = (a an? 4 (b +§c)1:
We have {(1, é., , 0:)T, (1, 0, , 1)T}, a
MAT247S, 2009 Winter, Problem Set 6 Solution
Grader: TAM, Kam Fai, geo.tam@utoronto.ca
l
i1 (x) is (T I)-invariant and I1.(a) Since W =
i=1 F (T I)
invariant, it is also T -invariant.
(b) By denition (T I)l (x) = 0, and (1)l (t )l is of deg= l = dim(W ).
MAT247 - Problem Set 1 Solution
Problem 1
a) It is straightforward to verify this is an inner product.
b) Again, easily veried using matrix multiplication.
c) Not an inner product. We have < x, x >= x2 4x2 , take x1 = 5 and
1
2
x2 = 1, we see that < , > i
MAT 247S - Problem Set 4
Solution to question 5
5. Let V be a nite-dimensional real inner product space and let m be a positive integer. Suppose
that T L(V ), T is normal, and T m = T0 (that is, T m (x) = 0 for all x V ). Prove that
T = T0 .
Solution: Bec
MAT247S, 2009 Winter, Problem Set 3 Solution
Grader: TAM, Kam Fai, geo.tam@utoronto.ca
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
MAT247S, 2009 Winter, Problem Set 4 Solution
Grader: TAM, Kam Fai, geo.tam@utoronto.ca
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 5 Solution
Grader: TAM, Kam Fai, geo.tam@utoronto.ca
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(
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
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,
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,