Linear Algebra III
Math 3107, Fall 2012
Assignment 1.
Due date: Tuesday October 2 in tutorial.
Exercise 0.1. Let
cfw_(
) (
) (
)
2i
5 2i
6 4i
U = span
,
,
4 + 3i
5 + 3i
17 6i
(1) Consider U a subspace C U of C2 over C  that is, the
vector space of pairs
MATH3107
1. Given that A =
(a) B = [ 1 2 2 ]t
SAMPLE TEST
, solve the system AX = B when
(b) B = [ 0 2 2 ]t
(transposed for typographic convenience)
2. Find the LU decomposition for the matrix A =
.
3. Find a basis and the dimension of each of the follo
MATH3107
SAMPLE TEST2
1.Let T : M32 6 M23 be the mapping defined by
.
(i) Show that T is a linear transformation (just use the 1,1 element as a typical case).
(ii) Determine whether or not T is onetoone.
(iii) For the standard bases 1 and 2 for M32, M23
MATH3107
TEST2
Answers
1. S(a+bt+ct2 +aN+bNt+cNt2) =
= S(a+bt+ct2) + S(aN+bNt+cNt2) and S(d(a+bt+ct2) =
= dS(a+bt+ct2) and so S
is a linear transformation.
T(1+t2) =
, T(1) = T(t2) = 0. T(1) + T(t2) T(1+t2) so T is not a linear transformation.
(i) S(a+bt+
MATH3107
SAMPLE TEST3
1. Find the characteristic equation, the eigenvalues and corresponding eigenvectors for
the following matrices :
(i) the real matrix A =
(ii) the real matrix A =
(iii) the complex matrix A =
If possible, find a matrix P and a diagona
MATH3107
Sample Questions II
DISCLAIMER: These questions are only samples! They do not purport to cover everything
which may be on the final exam and may also cover topics more exhaustively than the actual
exam. Solving these problems is entirely at your
MQTH 3107 PART I
p 1.3
10) page M 40" P171)? a+01+m+09 ~(q+(,)"x? =a(+ 23 713)P:;7)_.
I; a(4+172'13')&('1+111g)Wmao=>zo sz(=>: O
W cfw_'44 2% 9(+1 13 W W DE; 8
c 00]
(a e Tm [22,1 [(331 m] [23:] QTL [338 Vmw z
(3)V3= N '\32,_ 4.04); M (WW 1;: amrkbv 1+c
February 17, 2011
1
TEST SOLUTIONSMATH 3107
1. Compute the standard form of the complex number
a + bi for some a, b R.)
2+i
23i
(i.e. write as
Solution:
2+i
2 3i
=
=
=
=
1
2 3i
2 + 3i
(2 + i)
4+9
4 3 + (6 + 2)i
13
1
8
+ i
13 13
(2 + i)
2. Compute the stan
MATH 3107 Linear Algebra III
Winter 2014
Lecture Notes. Vector Spaces and Subspaces.
1. Vector Spaces
Denition 1.1. A vector space is a nonempty set V of objects, called vectors, with
two operations, called addition and scalar multiplication, which satisf
September 4 (Friday) Health Care Systems:
Hurley, Chapter 1 pp. 124
Garber, Alan M. and Jonathan Skinner (2008), Is American health care uniquely inefficient?, Journal of
Economic Perspectives 22(4), 2750.
Hurley, Chapter 1 pp. 124
Chapter 1: Health an
MATH3107
SAMPLE TEST2
Answers
1.(i) For matrices A and AN, the (1,1) entry of T(A+AN) is a+aN+b+bN = a+b+aN+bN which
is the (1,1) entry of T(A)+T(AN). For T(cA) it is c(a+b) = ca + cb which is the entry for
cT(A). All other entries follow so T is linear.
MATH3107
1.(i) A  I =
TEST3
Answers
= 4  8 + 52  3 so the C.E. is 3  52 + 5  4 = 0.
This factors to (  2)2(  1) so = 2, 2, 1.
= 2 gives x 1 + x 3 = x 2 with corresponding eigenvectors [1 1 0]t and [0 1 1]t.
= 1 gives x 1 = x 2 = x 3 and an eige
Linear Algebra III
Math 3107, Fall 2012
Assignment 3. Solution.
Exercise 1. Let A be an n n matrix. Are the following
statements true or false?
(1) If A is diagonalizable then A has n distinct eigenvalues.
(2) If AP = P D for some diagonal matrix D then t
MATH 3107 Linear Algebra III
Fall 2012
Lecture Notes (brief and incomplete). November.
Contents
1. Matrices with real entries
1.1. Orthogonal Matrices
1.2. Orthogonal Diagonalization of Symmetric Matrices
1.3. Positive Denite Matrices
2. Quadratic Forms
3
Jiayu Lin
Assignment ProblemSet4 due 10/25/2012 at 07:59pm EDT
1. (1 pt) Given:
2
6
T(
)=
2
20
6
57
T(
)=
7
8
Find a matrix such that:
T (v)=
2012FALLMATH3107
5. (1 pt) Let V be a vector space, and T : V V a linear transformation such that T (5v1 3v2
Jiayu Lin
Assignment ProblemSet3 due 10/07/2012 at 04:59pm EDT
2012FALLMATH3107
5. (1 pt) Which of the following transformations are linear?
1. (1 pt) (a) If S is the subspace of M3 (R) consisting of all
lower triangular matrices, then dim S =
(b) If S i
Jiayu Lin
Assignment ProblemSet2 due 10/07/2012 at 04:55pm EDT
2012FALLMATH3107
F. cfw_p(t)  p (t) is constant
1. (1 pt) Which of the following subsets of R33 are subspaces of R33 ?
Answer(s) submitted:
(incorrect)
A. The 3 3 matrices whose entries a
Linear Algebra III
Math 3107, Fall 2012
Assignment 1. Solution.
Exercise 1. Let
cfw_(
) (
) (
)
2i
5 2i
6 4i
U = span
,
,
4 + 3i
5 + 3i
17 6i
(1) Consider U a subspace C U of C2 over C  that is, the
vector space of pairs of complex numbers with scalars
f
Linear Algebra III
Math 3107, Fall 2012
Assignment 2. Solution.
Exercise 1. Compute det T for the following transformations:
(1) T : R C R C dened by T (z) = z (complex conjugation), where R C
is the real space C.
Solution. Here and below we will simply w
Linear Algebra III
Math 3107, Fall 2012
Assignment 2.
Due date: Tuesday October 30 in tutorial.
Exercise 1. Compute det T for the following transformations:
(1) T : R C R C dened by T (z) = z (complex conjugation), where R C is the real space C.
p(x 1).
(
MATH3107
PART I (Only the answers will be marked.)
Question 1. [7 marks] Exhibit a basis and calculate the dimension of each of the following
REAL spaces:
(1) The subspace U = cfw_a + bx + (a+b)x 2 + cx 3  a + b + c = 0, a, b, c 0 of P3 .
(2) The space

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