132 ' CHAPTERZ Matrix Algebra
PRACTICE 'PsoeL-EMs
1. Since vectors in R” may be regarded as n x 1 matrices, the properties of transposes
in Theorem 3 apply to vectors, too. Let '
4+: ii and we
Compute (Ax)T, x13“, HT, and XTX. Is ATXT deﬁned?
- 2. Let A b
Eigenvalues and Eigenvectors
PRACTICE PROBLEMS
6w3
1. IsSaneigenvalueofA: 3 0 5 ‘2
2 2 6
2. If x is an eigenvector for A 'corresponding to l, what is A3x?
. 1 EXERCISES
1. Is A. = 2 an eigenvalue of ? Why or why net?
2. Is A = #2 an eigenvalue of
3. Is a
3'98 _ CHAPTER 6 Orthogonality and Least Squares
The veriﬁcation for R3 is similar. When n > 3, formula‘(2) may be used to
angle between two vectors in IR”. In statistics, for instance, the value of cos
by (2) for suitable vectors u and v is what statisti
206 CHAPTER 3 Determinants
orks extremely well when an entire
le 3 of looking for zeros w
The strategy in Examp
row or .column consists of zeros. In such a case, the cofactor expansion along such arow
or column is a sum
Determinant
Determinantisdefinedandonlydefinedforsquarematrices.
Theorem.Let beafunctionof columnvectors ,.,
linearandskewsymmetricinthese columnvectors,i.e.,
in
.Supposethat ismulti
and
Then iscompletelydeterminedbytherealnumber
vectorwhose thentryis and
So at most two rows of A have a pivot.
The result does not depend on the order of the two
operations (I.e., matrix multiplication and vector addition)
The result does not depend on the order of the two operations
(I.e., matrix multiplication and scalar mu
Parallelogram rule for vector addition
Question: is a linear combination of
?
Answer:
b is a linear combination of
matrix
is non-pivot.
<=> the last column of
the domain
the image of
the codomain
under T
is linear means that T commutes with (the
operation of taking) linear combination. I.e.,
linear transformations from
to
matrices
Section 1.9 (Through Theorem 10) The Matrix of a Linear
Transformation
Identity Matrix In is an n x n matrix with 1’s on the main left to
right diagonal and 0’s elsewhere. The ith column of I,l is labeled
ei.
EXAMPLE:
Note that
= O + {a ‘ + 0 : 2g
0 o l
l
HKUST MATH2121 Linear Algebra (Fall 2015)
Final Examination
Name:
Student I.D.:
09 Dec 2015
Students Signature:
DIRECTIONS:
Do NOT open the exam until instructed to do
so.
All mobile phones and pagers should be
switched OFF during the examination.
You
MATH 2131
These notes are a
preliminary
typeset version of the handwritten
webnotes. It is meant to be the handwritten notes in a nice format. However, many examples in the handwritten have been omitted because of the
effort it would take to typeset them.
Problem under Hoftstedes cultural dimension theory
Aspect:
A. Power distance
Attitude: 1. staff dont recognize Fitchs work.
B. Unceartainty Avoidance
Orginazational problem(OG)
1. Overbook
2. lack of proper shift, misplacement of staff
3.Lack of specifica
Current Situation:
Investment Choice: Conservative Fund +: Risk free
-: Low Return(current VS expected figure)
Monthly Budget
Net Worth
Saving(2.5%) liquidity?
Current Annuity amount, pay rate
Asset Allocation Philosophy:
Growth in portfolioSwitch from co
MATH 2121
Fall 2012
Assignment Solution: Module VIII
VIII. Coordinates, Matrix representations
VIII.1.
a
b
Let V be the vector space of 2 2 matrices over and let M
.
c d
Find the matrix representation [T ] of the linear transformation T ,
T ( A) MA. ,on
Assignment Solution
MATH 2121
Fall 2012
Linear Transformation
VII. Linear Transformation
VII.1.
Let F : V U be linear. Show that (i) F (V ) cfw_F (v ) : v V ,the image of
any subspace V of V , is a subspace of U and (ii)
F 1 (W ) cfw_v V : F (v) W , the
Assignment Solution
MATH 2121
Fall 2012
X: Eigenvalues, Eigenvectors Diagonalizations
X.1.
Show that similar matrices have the same eigenvalues.
X.1. Solution:
Let A and B be similar. Then, there exists an invertible matrix P
such that B P 1 AP.
Let
Av v,
Assignment Solution
Determinants
MATH 2121
Fall 2012
IX. Determinants
IX.1
Find det(T ) for the linear transformation T :
T is the linear transformation on the vector space V of 2 2 matrices
over defined by
a b
T ( A) MA where M
c d
Hint: Use the follow
Assignment Solution
V
MATH 2121
Fall 2012
V. Vector Spaces and Subspaces
V.1.
Let U and W be vector spaces. Let V be the set of ordered pairs (u, w)
where u belongs to U and w to W : V u, w : u U , w W . Show that
V is a vector space with addition in V an
Assignment Solution
MATH 2121
Fall 2012
VI. Linear Independence, Basis and Dimension
VI.1.
Suppose cfw_u , v, w is a set of linearly independent vectors. Show that
cfw_u v, u v, u v w is linearly independent.
VI 1 solution:
Given a linearly independent se
Exercise : Use the properties a through h to show that 1) 0 A = 0, 2) (-1)A = -A.
commutivity:
associtivity:
existence of zero:
existence of
negative:
g. For each matrix A, there is
a matrix B such that A+B = 0.
We write B as -A.
h. 1 A = A
<-> compositio
5.3
Diagonalization
The goal here is to develop a useful factorization A = PDP 1 ,
when A is n n. We can use this to compute A k quickly for large
k.
The matrix D is a diagonal matrix (i.e. entries off the main
diagonal are all zeros).
D k is trivial to c
Section 5.1
Eigenvectors & Eigenvalues
The basic concepts presented here - eigenvectors and
eigenvalues - are useful throughout pure and applied
mathematics. Eigenvalues are also used to study difference
equations and continuous dynamical systems. They pr
Exercise: Show that (AB) ) = (A ) (B ).
elementary row operations on m x n-matrices
elementary matrices of order m
elementary column operations on p x m-matrices
Facts: If A is a mxn-matrix, then
1) A = PA where A is in row reduced echelon form and P is a
Proof. 1) Assume that S is a linearly independent set. Then there are numbers c , ., c ,
not all equal to zero, such that c v + . + c v = 0. Without loss of generality, we may
assume that c = 0 . Then
V=
so
is a linear combination of
2) Assume one of the
Section 5.2 The Characteristic Equation
Review:
A
x
=
x
Find eigenvectors x by solving A I x = 0.
How do we find the eigenvalues ?
x must be nonzero
A I x = 0 must have nontrivial solutions
A I is not invertible
detA I = 0
(called the characteristic equat
4.2 Null Spaces, Column Spaces, & Linear Transformations
The null space of an m n matrix A, written as Nul A, is the set
of all solutions to the homogeneous equation Ax = 0.
Nul A =
x : x is in R n and Ax = 0
(set notation)
THEOREM 2
The null space of an