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Matrix Multiplication
matrix multiplication provides a useful way of using vectors
and matrices to represent systems of equations
next we'll dene matrix multiplication and show how it can be
applied
Systems of equations in matrix form
Example: consider th
Leontie Model (example)
consider an economy that produces three things:
food (good 1)
raw materials (good 2)
manufactured goods (good 3)
suppose that workers supply labour inelastically, but are paid in
food and manufactured goods
production of each good
Multivariable Calculus: a sample problem
Firm input and output choice
Consider a rm facing a downward sloping demand curve. The rm
uses n inputs to produce output, and can choose among dierent
combinations of inputs.
If the rm maximizes prots, how will it
Mathematical Thinking
mathematical models force us to think logically based clearly
I
but: need to apply mathematical reasoning correctly!
Example
Problem
Find solutions to the equation x + 2 =
p
4
x.
Example
Problem
Find solutions to the equation x + 2 =
Vectors
vectors are a convenient way of denoting a multidimensional
quantities
e.g. consumption bundle
dened a vector as a matrix with either only one row or only
one column
can instead think of a vector in n-dimensional Euclidean space
Rn
then, the row/c
Claim:
If a and b are vectors in R2 , then |a b| a b .
Proof:
Since |a b|, a b 0, it is equivalent to show that a 2 b 2 |a b|2 .
If a = 0 or b = 0, then both sides of the expression are equal to zero and
the expression holds.
Now suppose a = 0.
Dene c = (
Homework note (for Nov 18).
(i) Since ux , uy , uz > 0 on the domain of u, u is strictly increasing. If (x, y, z) is
a bundle, and px x + py y + pz z < I , then, the bundle (x + px , y + py , z + pz ) also
satises he budget constraint whenever (0, I ]. Si
Linda is 31 years old, single, outspoken, and very bright.
As a student, she was deeply concerned with issues of
discrimination and social justice, and also participated in
anti-nuclear demonstrations.
Linda is 31 years old, single, outspoken, and very br
Gaussian Elimination
how can we solve systems of equations?
e.g. multimarket supply and demand with complements and
substitutes
Gaussian elimination: an approach to solving linear systems of
equations with many unknowns
System of Equations: Example
x1 + x