1
AM 1411a, Fall 2014
Lectures 1-5,
September 5-15
Ex. Tra c ow (equally well, it could be ow of water
in a pipe network)
The numbers on Figure are rates of ow: cars/minute
(or litres/minutes for ow of water in pipes). There are
four unknown currents: i1;

1 AM 1411a, Fall 2014
Lectures 20, October 22
Linear independence (Sec. 4.2) Recall(p.139): Def 1. If W is a vector in V, then W is
called a linear combination of the vectors V1, V2, ., VT
in V, if it can be expressed as
W zklvl + [2V2 l l krV'r
where k1,

1
AM 1411a, Fall 2014
Lectures 28, November 12
Similarity (Sec. 5.2)
Def. If A and B are square matrices, then they are called
similar matrices if there is an invertible matrix P such that
B = P 1AP (or A = Q 1BQ, where Q = P 1).
Similarity invariants. B

1
AM 1411a, Fall 2014
Lectures 29, November 14
Orthogonal matrices (Sec. 7.1)
Def 1. A square matrix A is said to be orthogonal if
A 1 = AT :
A 1A = AA 1 = I ) AT A = AAT = I
Def 2: An n n matrix A is said to be orthogonal if
the row vectors and the colum

1 AM 14118. Fall 2014
Lectures 21, October 24
Linear independence (Secs. 4.2-4.3)
Ex 1. One more important subspace: AX : 0 of m
equations in n unknown is a subspace W of B.
Let Xland x2 are solutions: Axl = O and AX2 : 0.
2) AUch) 2 Ich1 2 k0 = 0
3) Zero

1
AM 1411a, Fall 2014
Lectures 16, October 10
Properties of determinants
Th. 2.2.4. Let E be an n
(a) scaling (k 6= 0): I
n elementary matrix.
kRi ! Ri E , then
!
det (E ) = k:
(b) interchange: I
(c) elimination: I
det (E ) = 1:
Ri $ Rj E , then det (E )

1
AM 1411a, Fall 2014
Lectures 23, October 29
Dimension (Sec. 4.5)
Def. The dimension of V; dim (V ) ; is the number of
vectors in a basis for V .
The zero vector has dimension zero: dim (f0g) = 0:
dim (Rn) = n; dim (Mm n) = mn
dim (Pn) = n + 1 (polynomia

1
AM 1411a, Fall 2014
Lectures 24, November 3
Coordinates (Sec. 4.4)
Def. If S = fv1; v2; :; vng is a basis for a vector space
V , and
v = c1v1 + c2v2 + : + cnvn
is the expression for a vector v in therm of the basis S ,
then the vector
(v)S = (c1;c2;:; c

1
AM 1411a, Fall 2014
Lectures 25, November 5
The fundamental matrix spaces
(Secs. 4.7-4.8)
The fundamental matrix spaces:
row (A) ; col (A) ; null (A) ; null AT :
The basis for row (A) is (all non-zero rows in REF of A);
The basis for col (A) is (all col

Lectures 22, October 27
Basis and dimension (Secs. 4.44.5)
Def. if S : {V1,V2,.,V7«} is a set of vectors in a
vector space V, then S is called a basis for V if:
(a) S spans V;
(b) Sis LI.
Ex 1. R2 Ex 2. Checking for a basis
bl
(a) S spans R3 : let b 2 [

1
AM 1411a, Fall 2014
Lectures 8, September 22
A homogeneous system of linear equations
a11x1 + a12x2 + : + a1nxn = 0
a21x1 + a22x2 + : + a2nxn = 0
:
am1x1 + am2x2 + : + amnxn = 0
is always consistent;
a) either it has a unique solution: x1 = x2 = : = xn

Exercises for AM1411
D.J. Jerey
Department of Applied Mathematics
U. Western Ontario
February 10, 2009
1
Elementary Matrices
1. A matrix A = [aij ] is dened to be diagonal if
(a) the elements obey aij = 0 for all valid i, j .
(b) the elements obey aii = 0

Exercises for AM025
D.J. Jerey
Department of Applied Mathematics
U. Western Ontario
March 15, 2007
1
Vectors and Transformations
1. Plot the image of a square under the following transformation, and show it is a shear. (The area is unchanged.)
y1
y2
1
0
=

1 AM 14113, Fall 2014
Lectures 19, October 20
General Vector Spaces (Sec. 4.1) Def 1. (p. 184) Let V be a set of objects on which
two operations are defined: addition (11 l v) and scalar
multiplication where u and V belong to V and k
is a scalar. If the f

1
AM 1411a, Fall 2014
Lectures 18, October 17
Euclidean Vector Spaces
(review Secs. 3.1, 3.2, 3.3 (orthogonality)
Def 1. If n is a positive integer, then an n-tuple is called
a vector ! = v = (v1; v2; :; vn). The set of all nv
vectors is called the Euclid

1
AM 1411a, Fall 2014
Lectures 12, October 1
An algorithm for nding
I ! E and E 1 ! I
elem. row op.
A 1
E ! I and I ! E 1
inverse row op.
Theorem1.5.3. Equivalent statements. An n
(a) A is invertible.
(b) Ax = 0 has only the trivial solution.
(c) RREF of

1
AM 1411a, Fall 2014
Lectures 6, September 17
Ohm Law
s
V = IR
(R is a resistance, I is a current passes through the resistor, V is a voltage drop on the resistor.
Kirchho Current Law. For any node (or junction)
s
Iin = Iout
Kirchho Voltage Law. For any

1
AM 1411a, Fall 2014
Lectures 17, October 15
Inverse using the adjoint
Def. If A is any n n matrix and Cij is the cofactor of
entry aij , then the matrix
2
6
6
6
4
3
C11 C12 : C1n
C21 C22 : C2n 7
7
7 is called the
:
:
: 5
Cn1 Cn2 : Cnn
matrix of cofactor

1 AM 1411a, Fall 2014
Lectures 10, September 26
Inverses AI :2 1: AeA
Def. If A is a square matrix, and if a matrix B of the
same size can be found such that AB 2 BA : I, then
A is invertible (or nonsingular) and B is the inverse
B E Al. If no such matrix

1
AM 1411a, Fall 2014
Lectures 11, September 29
Elementary Matrices
Def 1. Matrices A and B are row equivalent if either
can be obtained from the other by a sequence of elementary row operation.
Def 2. A matrix E is called an elementary matrix if it
can b

1
AM 1411a, Fall 2014
Lectures 7, September 19
1.1
Polynomial Interpolation
Ex 1. Find an equation of the parabola that passes
through the points ( 1; 6), (1; 2) and (2; 9).
Solution
Let y = a + bx + cx2; then
point ( 1; 6) :
6=a
b+c
point (1; 2) :
2=a+b+

1
AM 1411a, Fall 2014
Lectures 8, September 22
Matrices and matrix operations
h
i
Examples of matrices: A = aij =
h
i
2
1
3
6
7
B = bij = 4 2 5 ; C =
3
h
"
i
1 2
3 4
#
1 2 3 ;D =
;
h
2
i
1. A = B , if aij = bij . (A and B have the same size.)
2. Sum of ma

1
AM 1411a, Fall 2014
Lectures 9, September 24
Matrix operations
A square matrix of order n is the matrix
2
6
6
A=6
4
a11 a12
a21 a22
:
:
an1 an2
3
: a1n
: a2n 7
7
7
:
: 5
: ann
Entries a11; a22; :; ann are on the main diagonal of A.
The trace of A is: tr

1
AM 1411a, Fall 2014
Lectures 9, September 24
Properties of matrices
Theorem 1.4.1. (p. 39)
(a) A + B = B + A (commutativity) (A and B are
of the same size)
.
(m) a (BC )= (aB ) C = B (aC ) where a is a scalar.
In contrast to operations with numbers, in