Download Theorem Ax = b is consistent if and only if b is a linear

Chapter 1. Matrices and Systems of Equations
Matrix Algebra
Theorem
Math1111
Consistency
Ax = b is consistent if and only if b is a linear
combination of column vectors of A.
Chapter 1. Matrices and Systems of Equations
Matrix Algebra
Theorem
Math1111
Consistency
Ax = b is consistent if and only if b is a linear
combination of column vectors of A.
Proof. “if”:
Write A = (a1
a2 · · · an ). Suppose b is a linear combination of
a1 , a2 , · · · , an . It means that we have some scalars α1 , α2 , · · · , αn
such that
α1 a1 + α2 a2 + · · · + αn an = b.
Chapter 1. Matrices and Systems of Equations
Math1111
Matrix Algebra
Theorem
Consistency
Ax = b is consistent if and only if b is a linear
combination of column vectors of A.
Proof. “if”:
Write A = (a1
a2 · · · an ). Suppose b is a linear combination of
a1 , a2 , · · · , an . It means that we have some scalars α1 , α2 , · · · , αn
such that
α1 a1 + α2 a2 + · · · + αn an = b.

Take

 α1 
 
α 
 2
x =  . ,
 .. 
 
 
αn
then Ax = b. i.e. The equation Ax = b
has solution.
Chapter 1. Matrices and Systems of Equations
Math1111
Matrix Algebra
Consistency (Cont’d)
Proof. “only if”:




a1 
a1 
 
 
a 
a 
 2
 2
Let  .  be a solution of Ax = b. i.e. A  .  = b.
 .. 
 .. 
 
 
 
 
an
an
Chapter 1. Matrices and Systems of Equations
Math1111
Matrix Algebra
Consistency (Cont’d)
Proof. “only if”:




a1 
a1 
 
 
a 
a 
 2
 2
Let  .  be a solution of Ax = b. i.e. A  .  = b.
 .. 
 .. 
 
 
 
 
an
an
It can be rewritten as
a1 a1 + a2 a2 + · · · + an an = b
where aj is the jth column of A. This means by definition that b
is a linear combination of the column vectors of A.