Download The Matrix Equation A x = b (9/17/04)

The Matrix Equation A x = b
(9/16/05)

Definition. If A is an m  n matrix with
columns a1, a2,…, an and x is a vector
in Rn, then the product of A and x,
denoted A x , is the linear combination
of the columns of A using the
corresponding entries of x as the
weights.
Three ways to view things:

Theorem. If A is an m  n matrix with
columns a1, a2,…, an and b is a vector
in Rm, the matrix equation A x = b
has the same solution set as the vector
equation x1 a1 + x2 a2 +…+ xn an = b ,
which in turn has the same solution set
as the system of linear equations with
augmented matrix [a1 a2 … an b]
Existence of Solutions

Theorem. If A is an m  n matrix. Then the
following statements are equivalent (i.e., for a
given A, either they are all true, or they are
all false):
m
 For each b in R , the equation A x = b
has a solution x.
m
 Each b in R is a linear combination of the
columns of A.
m
 The columns of A span R .
 A has a pivot position in every row.
The row-vector rule
for computing A x


Though the definition of the product of
an m  n matrix A and a vector x of
length n is made in terms of scalars and
vectors (see first slide), the computation
is simply that each row of the answer is
the sum of the products of the entries of
that row of A and the entries of x.
Check this idea by an example….
Arithmetic Properties

It is true, and easy to check by some
examples, that if A is an m  n matrix,
u and v are vectors in Rn, and c is a
scalar, then:


A (u + v) = A u + A v , and
A (c u) = c (A u) .
Assignment


On Monday, we will meet in the MCS lab
(Harder 209) and do some machine
computations of the row reduction
algorithm. Review that algorithm again
in preparation.
For next Wednesday, please:


Read Section 1.4.
Do the Practice and Exercises 1-19 odd and
23.