Then the two compositions are
BA =
10
01
=
0 1
10
AB =
Algebra of linear transformations and
matrices
Math 130 Linear Algebra
0 1
10
10
0 1
0 1
0 1
=
10
1 0
The products arent the same.
You can perform these on physical objects. Take
Weve looked at the op
where A and B are arbitrary constants.
Both of these equations are homogeneous linear
dierential equations with constant coecients. An
nth order linear dierential equation is of the form
Linear Dierential Equations
Math 130 Linear Algebra
an f n (t) + +
Linear transformations and matrices
Math 130 Linear Algebra
D Joyce, Fall 2013
One of the principles of modern mathematics is that functions between objects are as
important as the objects themselves. The objects were looking at are vector spaces, and the
a eld, except that the scalars are taken in the ring
R instead of in a eld.
For example, R[x] R[x] is a module over the
ring R[x].
The axioms for modules over Z can be simplied
since scalar multiplication by elements of Z can be
reduced to addition. For e
Notes on Matlab
Math 130 Linear Algebra
D Joyce, Fall 2013
What is Matlab? Its an interactive system for doing mathematics. It can perform tasks
that you could do by hand, but it wont make the mistakes we make and it works much faster.
You can ask it to d
Notes on Quaternions
Math 130 Linear Algebra
D Joyce, Fall 2013
Sir William Rowan Hamilton, who early found that his road [to success with vectors] was obstructedhe knew not by what obstacleso that many points which
seemed within his reach were really ina
Some linear transformations on R2
Math 130 Linear Algebra
D Joyce, Fall 2013
Lets look at some some linear transformations on the plane R2 . Well look at several
kinds of operators on R2 including reections, rotations, scalings, and others.
Well illustrat
For the time being, well look at ranks and nullity
of transformations. Well come back to these topics
again when we interpret our results for matrices.
The above theorem implies this corollary.
Kernel, image, nullity, and rank
Math 130 Linear Algebra
T
U
We can translate this as a theorem on matrices
where the matrix A represents the transformation
T.
Kernel, image, nullity, and rank
continued Math 130 Linear Algebra
Theorem 2 (Dimension theorem for matrices).
For an m n matrix A
D Joyce, Fall 2013
n = ra
get away with and so forth. If you want to see
all of what and so forth entails, you can read
Dedekinds 1888 paper Was sind und was sollen
die Zahlen? and my comments on it. In that article he starts o developing set theory and ends up
with the natural nu
Theorem 2. The span of a set S is a subspace of
V.
You can also describe span(S) as the smallest
subspace of V that contains all of S.
Span and independence
Math 130 Linear Algebra
Theorem 3. The span of a set S is the intersection
of all subspaces of V t
This second characterization is equivalent to the
rst because, rst, linear combinations are built
from vector additions and scalar products, and, second, scalar products and vector additions are special cases of linear combinations.
Subspaces of Vector Sp
Vectors
Math 130 Linear Algebra
D Joyce, Fall 2013
Welcome to the course! Much of today well discuss the administration of the course.
The course web page is at
http:/math.clarku.edu/~djoyce/ma130/
We will discuss the general description of the course, it
Theres also a choice of which operations on vector spaces to axiomatize. Here we used addition
and scalar multiplication. Alternatively, subtraction and scalar multiplication could be used. Addition was used because, in some sense, addition is
a more basi
For the time being, think of the scalar eld F as
being the eld R of real numbers.
The precise denition. A vector space over a
scalar eld F is dened to be a set V , whose elements we will call vectors, equipped with two operations, the rst called vector ad
'$ , sin )
q u=(x,y)=(cos
U
Applications of inner products in Rn
Math 130 Linear Algebra
&%
D Joyce, Fall 2013
Figure 1: Unit vectors in R2
Summary of norms and inner products in Rn discussed last time.
will land on the unit circle x2 + y 2 = 1. Every poi
Furthermore, the triangle inequality for complex
norms holds
vw v + w .
n
Norm and inner products in C ,
and abstract inner product spaces
Math 130 Linear Algebra
Well prove it later.
The inner product vw of two complex vectors. We would like to have a c
In many ways, norms act like absolute values.
For instance, the norm of any vector is nonnegative,
and the only vector with norm 0 is the 0 vector.
Like absolute values, norms are multiplicative in
the sense that
Norm and inner products in Rn
Math 130 Lin
Cauchys inequality
Math 130 Linear Algebra
D Joyce, Fall 2013
The triangle inequality and angles in nspace. We worked from principles of geometry
to develop the triangle inequality in dimension 2, and it works in dimension 3 as well, but we
cant rely on
(b1j , b2j , . . . , bnj ), so
b1j
b2j
[bj ] = .
.
.
Change of coordinates
Math 130 Linear Algebra
bnj
D Joyce, Fall 2013
Collect these in the columns
The coordinates of a vector v in a vector space form a transition matrix.
V with respect to a basis =
B
Fp

Fn
@
@
@
AB
A
@
R
@
Composition of linear transformations
and matrix multiplication
Math 130 Linear Algebra
?
Fm
Lets see what the entries in the matrix product
AB have to be.
D Joyce, Fall 2013
Let v be a vector in F p , then w = T (v) is a
vector
There are no real roots of this polynomial 2 + 1,
only the imaginary roots i. Thus this rotation
has no real eigenvalues and no real eigenvectors.
How can we continue on? We can treat the maRotations and complex eigenvalues
trix as a matrix over the compl
Although an isomorphism doesnt mean that V
is identical to F n , it does mean that coordinates
work exactly the same.
One place where this can be confusing is when
V is F n , but the basis is not the standard basis. Having two bases for the same vector sp
Cross products
Math 130 Linear Algebra
D Joyce, Fall 2013
The denition of cross products. The cross product : R3 R3 R3 is an operation
that takes two vectors u and v in space and determines another vector u v in space. (Cross
products are sometimes called
There are six terms in this determinant. Each term
is a product of three elements, one element chosen
out of each row and column. All six possible ways of
choosing one element out of each row and column
are included. Three have minus signs and three
have
Well see soon that if a linear operator on an ndimensional space has n distinct eigenvalues, then
its diagonalizable. But rst, a preliminary theorem.
Diagonalizable operators
Math 130 Linear Algebra
Theorem 2. Eigenvectors that are associated to
distinct
Other properties of determinants. There are
several other important properties of determinants.
For instance, determinants can be evaluated by
cofactor expansion along any row, not just the rst
row as we used to construct the determinant. We
wont take the
Determinants, part II
Math 130 Linear Algebra
D Joyce, Fall 2013
So far weve only dened determinants of 2 2 and 3 3 matrices. The 2 2 determinants
had 2 terms, while the determinants had 6 terms.
There are many ways that general n n determinants can be de
Proof. Let 0 = cfw_b1 , b2 , . . . , bn be a basis of a
vector space V , and let T = cfw_w1 , w2 , . . . , wr be a
set of linearly independent vectors in V . We need
to show that r n.
The idea is to replace the vectors in 0 by vectors
in T , one at a ti