Motivation
2.1 Determinant
of a Matrix
Suppose A =
a b
c d
Finding the Determinant of a 3 3 matrix
a11 a12 a13
Let A = a21 a22 a23
a31 a32 a33
One
Approach
Sarruss Rule for 3 3 matrices
Example
Find the determinant of
1 5
0
A = 2 4 1
0 2 0
Notation
deter

1.3 Matrix Arithmetic
Vectors
Motivation:
Ordered pairs (x, y) = (1, 2)
Ordered triplets (x, y, z)
n ordered items, n-tuples
(x1 , x2 , , xn )
allows us to address a large range of problems.
Q/A
What is the difference between a point and a vector?
Colu

3.3 Linear Independence
A homogeneous system such as
1 2 3
x1
0
3 5
9
x2 = 0
5 9
3
x3
0
can be viewed as a vector equation
1
2
3
0
9 = 0 .
x1 3 + x2 5 + x3
5
9
3
0
The vector equation has the trivial solution (x1 = 0, x2 = 0, x3 = 0),
but is this

1.6 Partitioned Matrices
Recall
Example
Q/A
A matrix can be partitioned into rows and columns. More generally, a matrix
can be partitioned into submatrices or blocks.
2 1 0
1 2 3
3 1 2
AB =
4 5 6
0 1 1
Suppose A is so large that you cannot access the ac

3.6 Row Space and Column Space
Theorem
3.6.1
Two row equivalent matrices have the same row space.
Example
Find a basis for the row space, a basis for the column space, and a basis for
the null
space of
1 0 1 1
A = 1 1 2 3
1 1 0 1
Theorem
3.6.6
If A is a

Preliminaries: Sets and Complex Numbers
Sets
Definition
- Ricardo Appendix A.1
A set is a collection of objects, called elements.
We use
to indicate that the collection is a set.
Examples
The set of natural numbers:
The set of all odd, positive integers

2.2 Properties of Determinants
Recall
Computing the determinant for some matrices is easy!
Otherwise, use cofactor expansion.
Finding Determinants using Row Operations
Q/A
Example
a b
Let A =
, det(A) =
c d
c d
1. If B =
, det(B) =
a b
a b
2. If B =
,
c d

3.1 Vector Spaces
Much of the framework we have developed for dealing with vectors in Rn can
be extended to treat other mathematical objects.
We can think of a vector space as a collection of objects that behave as
vectors do in Rn . The objects of such a

1.5 Elementary Matrices
Recall
Row operations
Scaling - multiply a row by a nonzero constant
Interchange - switch two rows
Replacement - add multiple of one row to another row
Definition
3 Types
An elementary matrix is one that is obtained by performin

2.3 Additional Topics and Applications
Cramers Rule
Lemma 2.2.1
Let A be an n n matrix and Aij the cofactors, then
ai1 Aj1 + ai2 Aj2 + . + ain Ajn =
Definition
Example
A11 A21 An1
A12 A22 An2
The adjoint of A is adjA = .
.
.
.
A1n A2n Ann
2 1
3
Compu

Section 1.2: Row Echelon Form
Definition
A matrix is in echelon form (or row echelon form) if it has the following
three properties:
1. The first nonzero entry in each nonzero row is 1.
2. Each leading entry (i.e. left most nonzero entry) of a row is in a

Section 1.1: Systems of Linear Equations
A linear equation in n unknowns:
a1 x 1 + a2 x 2 + + an x n = b
where a1 , a2 , . . . an and b
Examples
x1 , x2 , .xn
Which equations are linear?
x2 = 2( 6 x1 ) + x3
4x1 5x2 + 2 = x1
4x1 6x2 = x1 x2
A Simple
System

1.4 Matrix Algebra
Motivation
Theorem
1.4.1
Algebraic rules for real numbers may or may not extend to matrices.
1 2
3 1
1 1
=
2 3
1 1
2 3
1 2
=
3 1
Let A, B, and C be matrices, and let and be scalars.
1.
2.
3.
4.
5.
Warning!
A+B=B+A
(A + B) + C = A + (B +

3.4 Basis and Dimension
Review
Span(v1 , v2 , , vn ) =
v1 , v2 , , vn are linearly independent if
c1 v1 + c2 v2 + + cn vn = 0
Column Space
Null Space
Definition
The vectors v1 , v2 , , vn form a basis for a vector space V if and only if
(i) v1 , v2 ,

3.2 Subspaces
Vector spaces formed from subsets of other vectors spaces are called subspaces.
Definition
A subspace of a vector space V is a subset S of V that has three properties:
a. The zero vector is in S.
b. For each x and y in S, x + y is in S.
(S i