Department of Mathematics
University of the Philippines-Diliman
Math 114 - TFV
J. Basilla
1. Let A =
1
1
First Long Examination
13 July 2007
2 3
1
and B =
0 2
2
5 2
Compute for
2 1
(a) (2 points) A + B.
Solution:
2 3
1
+
0 2
2
1
1
5 2
0
=
2 1
1
7
2
1
.
1
XIatir
llilst
11.1
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Math 114 Exercises Set 1
3 0
4
1
1
4
2
I. Consider the following matrices: A = 1 2 , B =
, C =
,
0
2
3 1 5
1 1
1 5 2
6 1 3
D = 1 0 1 , E = 1 1 2 .
3 2 4
4 1 3
Compute the following (whenever possible):
1. (2DT E)A
4. (BAT 2C)T
2. (4B)C + 2B
5. B T (CC T A
Math 114 Exercises Set 2
1. Let V be the set of all positive real numbers. Define the following operations:
i. u v = uv
ii. v = v
Prove that V is a vector space.
2. Explain why the following are not vector spaces.
a. The set of all integers Z with the fo
Math 114 Exercises Set 3
1. Which of the following functions are linear transformations?
a. L : R3 R3 defined by L(x, y, z) = (x, y 2 + z 2 , z 2 )
b. L : R3 R3 defined by L(x, y, z) = (0, z, y)
c. L : Mmn (R) Mmn (R) defined by L(A) = AT
d. L : Mmn (R) M
Math 114 Exercises Set 4
1. Let L : R3 R3 be defined by L(x, y, z) = (2x + 3y, y + 4z, 3z). Using the natural basis
for R3 , find the eigenvalues and associated eigenvectors of L.
2. Give the characteristic polynomial, the eigenvalues and associated eigen
1
The Column Space & Column Rank of a Matrix
E. L. Lady
0
Let A = 0
0
1
2
1
2
4
2
3
0
5
4
20 .
0
The column space of A is the subspace of R3 spanned by the columns of A ,
in other words it consists of all linear combinations of the columns of A :
4
3
Matrix Theory
4
Homework 2
June 29, 2009
1. (a) Describe a! subspace of Mat22 !(the space of all 2 2 real-valued matrices) that contains
1 0
0 0
A=
but not B =
0 0
0 1
!
1 0
(b) If a subspace of Mat22 contains A and B, must it contain the identity matrix
Gram-Schmidt orthogonalization
Gram-Schmidt method
Examples
Orthogonal matrices
QR decomposition
Least squares and Gram-Schmidt method
Suppose you have linearly independent vectors v1 , v2 , . . . , vn . How to find an
orthonormal basis for spancfw_v
The Trace of a Matrix
c
Copyright 2012
Dan Nettleton (Iowa State University)
Statistics 611
1 / 11
The trace of a square matrixnn
A = [aij ] is
trace(A) = tr(A) =
n
X
aii .
i=1
For example,
5
3 5
tr
4 1 2 = 5 1 + 7 = 11.
3
c
Copyright 2012
Dan Nettleton
MATH10212 Linear Algebra Brief lecture notes
48
Similarity and Diagonalization
Similar Matrices
Let A and B be n n matrices. We say that A is similar to B if there is an
invertible n n matrix P such that P 1 AP = B. If A is similar to B, we
write A B.
Rem
These notes closely follow the presentation of the material given in David C. Lays
textbook Linear Algebra and its Applications (3rd edition). These notes are intended
primarily for in-class presentation and should not be regarded as a substitute for
thor
Lesson 4
Definition and Properties of Determinants
One application of determinants is to determine whether a linear system with n equations in n unknowns has a
unique solution. We first define the concept of permutations before defining the determinants.
C+ Programming: From Problem Analysis to
Program Design, Fifth Edition
Chapter 2: Basic Elements of C+
Objectives
In this chapter, you will:
Become familiar with the basic
components of a C+ program, including
functions, special symbols, and identifiers
C+ Programming: From Problem
Analysis to Program Design, Fifth Edition
Chapter 5: Control Structures II
(Repetition)
Objectives
In this chapter, you will:
Learn about repetition (looping) control
structures
Explore how to construct and use countcontroll
Department of Mathematics
University of the Philippines-Diliman
Math 114 - TFV
J. Basilla
Second Long Examination
31 Aug 2007
1. Multiple Choice. Write the letter(in Upper Case) of the correct answer on the rst page of your blue
book.
(a) (1 point) Which
Institute of Mathematics
University of the Philippines-Diliman
Math 114 - X3
J. Basilla
Third Long Examination
15 May 2008
1. Let V be the vector space of all functions spanned by the functions cfw_1, sin x, cos x, x sin x, x cos x
V
and consider the expr
Department of Mathematics
University of the Philippines-Diliman
Math 114 - TFV
J. Basilla
Third Long Examination
6 Oct 2007
I. Write the letter(in Upper Case) of the correct answer on the rst page of your blue book.
(1) (1 point) Which of the following is
Institute of Mathematics
University of the Philippines-Diliman
Math 114 - X3
J. Basilla
1. Let A =
1
1
First Long Examination
23 April 2008
2 3
1
and B =
0 2
2
5 2
Compute for
2 1
(a) (2 points) A + B.
Solution:
1
1
2 3
1
+
0 2
2
5 2
0
=
2 1
1
7
2
1
.
1
(
Linear Transformations - Definition
and Examples
Definitions. Let V (with operations + and ) and W (with operations and ) be real vector
spaces. A function L : V W is called a linear transformation of V into W if
i. L(u + v) = L(u) L(v), for all u, v in V
Coordinates, Matrix of a Linear
Transformation and Change of Basis
Matrix
Recall: Let L : V W be a linear transformation. If cfw_v1 , v2 , . . . , vn is a basis for V and
v V , then L(v) is completely determined by the set cfw_L(v1 ), L(v2 ), . . . , L(v
Cofactor Expansion and
Applications of the Determinant
Definition. If A is a square matrix, then the (i, j)-minor of A denoted by Mij (A) (or simply
Mij ) is defined to be the determinant of the submatrix that remains after the ith row and jth
column are
Definition and Properties of
Determinants
One application of determinants is to determine whether a linear system with n equations
in n unknowns has a unique solution. We first define the concept of permutations before
defining the determinants.
Definitio
8. Diagonalization
E-mail: hogijung@hanyang.ac.kr
http:/web.yonsei.ac.kr/hgjung
8.1. Matrix Representations of Linear Transformations
Matrix of A Linear Operator with Respect to A Basis
We know that every linear transformation T: RnRm has an associated st