Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
INTRODUCTION
TO
LINEAR
ALGEBRA
Fourth Edition
MANUAL FOR INSTRUCTORS
Gilbert Strang
Massachusetts Institute of Technology
math.mit.edu/linearalgebra
web.mit.edu/18.06
video lectures: ocw.mit.edu
math.mit.edu/gs
www.wellesleycambridge.com
email: [email protected]
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
Supplement
7A
Rotational Invariance
In this supplement we show that the assumption of a central potential implies the conservation of angular momentum. We make use of invariance under rotations. The kinetic energy, which involves p2, is independent of th
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
8.1. Inner Product Spaces
Inner product
Linear functional
Adjoint
Assume F is a subfield of R or C.
Let V be a v.s. over F.
An inner product on V is a function
VxV > F i.e., a,b in V > (ab) in F s.t.
(a) (a+br)=(ar)+(br)
(b)( car)=c(ar)
(c )
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
8.4. Unitary Operators
Inner product preserving
V, W inner product spaces over F in R or C.
T:V > W.
T preserves inner products if (TaTb) = (ab)
for all a, b in V.
An isomorphism of V to W is a
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
7.4. Computations of Invariant
factors
Let A be nxn matrix with entries in F[x].
Goal: Find a method to compute the
invariant factors p1,pr.
Suppose A is the companion matrix of a
monic polynomial
p=xn+cn1xn1+c1x+c0.
# x 0 0 . 0
c0 &
%
(
1
x
0
.
0
c
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
Ch 4: Polynomials
Polynomials
Algebra
Polynomial ideals
Polynomial algebra
The purpose is to study linear
transformations. We look at polynomials
where the variable is substituted with
linear maps.
This will be the main idea of this book
to classify lin
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
6.3. Annihilating polynomials
CayleyHamilton theorem
Polynomials and transformations
(p+q)(T) = p(T)+q(T)
(pq)(T)=p(T)q(T)
Ann(T) = cfw_p in F[x] p(T)=0 is an ideal.
Proof:
p,q in Ann(T) k in F > p+kq(T)=0> p+kq in
Ann(T).
p in Ann(T), q in F[x] >
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
Polynomial Ideals
Euclidean algorithm
Multiplicity of roots
Ideals in F[x].
Euclidean algorithms
Lemma. f,d nonzero polynomials in
F[x]. deg d deg f. Then there exists a
polynomial g in F[x] s.t. either fdg=0 or
deg(fdg)<deg f.
Proof of lemma:
Theorem
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
Chapter 2: Vector spaces
Vector spaces, subspaces, basis,
dimension, coordinates, rowequivalence, computations
A vector space (V,F, +, .)
F a field
V a set (of objects called vectors)
Addition of vectors (commutative,
associative)
Scalar multiplicatio
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
Rowequivalences again
Row spaces bases
computational techniques using
rowequivalences
The row space of A is the span of row
vectors.
The row rank of A is the dimension of
the row space of A.
Theorem 9: Rowequivalent matrices
have the same row spaces
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
Chapter 1. Linear equations
Review of matrix theory
Fields
System of linear equations
Rowreduced echelon form
Invertible matrices
Fields
Field F, +,
F is a set. +:FxFF, :FxFF
x+y = y+x, x+(y+z)=(x+y)+z
unique 0 in F s.t. x+0=x
unique x s.t. x+(x) =
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
Chapter 3:
Linear transformations
Linear transformations, Algebra of
linear transformations, matrices,
dual spaces, double duals
Linear transformations
V, W vector spaces with same fields F.
Definition: T:VW s.t. T(ca+b)=c(Ta)+Tb
for all a,b in V. c in
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2012
3.6. Double dual
Dual of a dual space
Hyperspace
(V*)*=V* = ? (V is a v.s. over F.)
= V.
a in V. I: a > La:V*>F defined by
La(f)=f(a).
Example: V=R2. L(1,2)(f)= f(1,2)=a+2b, if
f(x,y)=ax+by.
Lemma: If a0, then La 0.
Proof: B=cfw_a1,an basis of V s
Korea Advanced Institute of Science and Technology
Linear algebra
PHY 103

Spring 2009
Introduction to Linear Algebra (MAS109): Final
2009. 5. 21 (Thu)
Instructor:
Seat Number:
7:00 PM 10:00 PM
Student Number:
Do not write in this box
Name:
Read the following instructions carefully before you start the examination.
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