DEFINITION:
A system of linear equations is said to
be homogeneous if it can be written in
the form Ax̄ = 0̄. Otherwise, it is nonhomogeneous.
EXAMPLE:
3x1 + 5x2 = 0
6x1 + 2x2 = 0
HOMOGENEOUS
3x1 + 5x2 = 1
6x1 + 2x2 = 0
NONHOMOGEN.
THEOREM:
Suppose the equation Ax̄ = b̄ is consistent for some given b̄, and let p̄ be a solution. Then the solution set of Ax̄ = b̄
is the set of all vectors of the form
w̄ = p̄ + v̄h,
where v̄h is any solution of the homogeneous equation Ax̄ = 0̄.
THEOREM:
Let A be an m × n matrix. Then the
following statements are logically equivalent:
1. A homogeneous system Ax̄ = 0̄ has
only the trivial solution.
2. There are no free variables.
3. Number of columns of A = Number
of pivot positions.
DEFINITION:
Vectors v̄1, . . . , v̄p are said to be linearly
dependent if there exist scalars c1, . . . , cp,
not all zero, such that
c1v̄1 + . . . + cpv̄p = 0̄.
Vectors v̄1, . . . , v̄p are said to be linearly
independent if the vector equation
c1v̄1 + . . . + cpv̄p = 0̄
has only the trivial solution.
EXAMPLE: Vectors


 
−2
1
v̄1 =  1  , v̄2 =  −1  ,
2
−4
are linearly dependent.
Vectors
 


1
−2
v̄1 =  1  , v̄2 =  −1  ,
2
−4
are linearly independent.
 
3
v̄3 =  5 
6
 
3
v̄3 =  5 
7
THEOREM:
Let A be an m × n matrix. Then the
following statements are logically equivalent:
1. A homogeneous system Ax̄ = 0̄ has
only the trivial solution.
2. There are no free variables.
3. Number of columns of A = Number
of pivot positions.
4. The columns of a matrix A are linearly independent.
THEOREM:
Let A be an m × n matrix. Then the
following statements are logically equivalent:
1. A homogeneous system Ax̄ = 0̄ has
a nontrivial solution.
2. There are free variables.
3. Number of columns of A > Number
of pivot positions.
4. The columns of a matrix A are linearly dependent.
5. At least one column of A is a linear
combination of other columns.
1. Let 





2. Let 





0
0
1
v̄1 =  0  , v̄2 =  1  , v̄3 =  0  .
1
0
0
(a) Are v̄1, v̄2, v̄3 linearly independent?
(b) Does {v̄1, v̄2, v̄3} span R3 ?
0
0
1
v̄1 =  0  , v̄2 =  1  , v̄3 =  2  .
0
0
0
(a) Are v̄1, v̄2, v̄3 linearly independent?
(b) Does {v̄1, v̄2, v̄3} span R3 ?
3. Let
 
 
 
0
0
0
1
v̄1 =  0  , v̄2 =  1  , v̄3 =  0  , v̄4 =  0  .
2
1
0
0
(a) Are v̄1, v̄2, v̄3, v̄4 linearly independent?
(b) Does {v̄1, v̄2, v̄3, v̄4} span R3 ?
4. Let




1
0
v̄1 =  0  , v̄2 =  0  .
0
1
(a) Are v̄1, v̄2 linearly independent?
(b) Does {v̄1, v̄2} span R3 ?
1. Let





2. Let





9
7
5
v̄1 =  0  , v̄2 =  2  , v̄3 =  4  .
−8
−6
0
(a) Are v̄1, v̄2, v̄3 linearly independent?
(b) Does {v̄1, v̄2, v̄3} span R3 ?
9
7
5
4.
v̄1 =  0  , v̄2 =  2  , v̄3 = 
−12
−6
0
(a) Are v̄1, v̄2, v̄3 linearly independent?
(b) Does {v̄1, v̄2, v̄3} span R3 ?
3.
 Let







5
9
9
7
4.
v̄1 =  0  , v̄2 =  2  , v̄3 =  4  , v̄4 = 
−12
−8
−6
0
(a) Are v̄1, v̄2, v̄3, v̄4 linearly independent?
(b) Does {v̄1, v̄2, v̄3, v̄4} span R3 ?
4. Let




7
5
v̄1 =  0  , v̄2 =  2  .
−6
0
(a) Are v̄1, v̄2 linearly independent?
(b) Does {v̄1, v̄2} span R3 ?