At the end of week zero, distribution of T = 20, 000:
How to solve linear systems of dierence equations
10, 000 = 0.5T
What is a Markov Chain?
8, 000 = 0.4T
2, 000 = 0.1T
Number of shoppers at the end of the next week?
Lecture 4 Elementary Matrices
When does A1 exist?
Elementary row operations:
RO1 multiply a row by a non-zero constant
Find A1 .
RO2 interchange two rows.
RO3 add a multiple of one row to another.
Suppose B is a 3 3 matrix, e.g.
1 0 1
Preliminary Review Question
Suppose the RREF of a matrix A has the form
What can you say about the system of equations Ax = b?
definition as a cofactor expansion
If A = (C|d), What are the solutions of Cx =
Last week two concepts: linear independence and span
The set cfw_v1 , v2 , , vr is a basis of a vector space V .
The set of vectors cfw_v1 , v2 , . . . , vk spans V
What does it mean?
Why do we need it?
V = Lincfw_v1 , v2 , . . . , vk = cfw_
Lecture 7 Vector Spaces
What is a vector space?
linear combinations in a vector space
Deciding if a subset is a subspace
Lecture 3 Linear Systems
Look at solutions of systems of equations, Ax = b
A system of linear equations
Look at homogeneous systems of equations,
Ax = b
Ax = 0
can have a unique solution
Discover the Principle of Linearity
can be inconsisten
Lecture 1 Matrices
What is a matrix?
matrix = rectangular array of numbers or symbols
2 1 7 8
A = 0 2 5 1
1 1 3 0
m n matrix m rows and n columns
aij = (i, j) entry = number in the ith row, jth column.
2 1 7