Dr. Byrne
Math 237
Practice Quiz 7A: Applying the Test for Linear
Dependence and Finding a Basis for The Span
1 
0 
0 
1 
0 
2




,v 
, v  
v1 
0  2  2  3  3
 
 
 
0 
 3
0 
1) TEST FOR LINEAR DEPENDENCE

Write down the dependency equation x1v1  x2v2  x3v3  0 that
is used to test whether the set of vectors v1 , v2 and v3 are
linearly dependent.
2) SOLVING THE DEPENDENCY EQUATION
Find the solution set to the dependency equation using
Wolfram-Alpha or Octave to row reduce the augmented matrix
of the associated linear system. Describe the solution set in
parametric form (that is, with a translation vector and possibly
spanning vectors).
3) INTERPRETING THE SOLUTION
What is the result of the test regarding the linear dependence
of v1 , v2 and v3 ?
4) IDENTIFYING A BASIS FOR THE SPAN
The span of v1 , v2 and v3 is the set of all linear combinations of
v1 , v2 and v3 v4 .Write down a basis for the span of v1 , v2 and v3 .
Practice Quiz 7B: Applying the Test for Linear Dependence
and Finding a Basis for The Span
1
2
  1
2
 3
 6
4
 5
,
,
,
v1    v 2    v3    v 4   
4
8
  6
6
 
 
 
 
  1
 2
0
 3
1) TEST FOR LINEAR DEPENDENCE

Write down the dependency equation x1v1  x2v2  x3v3  x4v4  0
that is used to test whether the set of vectors v1 , v 2 , v 3 and v4 are
linearly dependent.
2) SOLVING THE DEPENDENCY EQUATION
Find the solution set to the dependency equation using
Wolfram-Alpha or Octave to row reduce the augmented matrix
of the associated linear system. Describe the solution set in
parametric form (that is, with a translation vector and possibly
spanning vectors).
3) INTERPRETING THE SOLUTION
What is the result of the test regarding the linear dependence
of v1 , v 2 , v 3 and v4 ?
4) IDENTIFYING A BASIS FOR THE SPAN
The span of v1 , v 2 , v 3 and v4 is the set of all linear combinations
of v1 , v 2 , v 3 and v4 .Write down a basis for the span of v1 , v 2 , v 3
and v4 .
Extra: Describe all vectors v1 , v 2 , v 3 and v4 as a linear
combination of vectors in the given basis.
v1 =
v2 =
v3 =
v4 =
Redoing 7A Using the Row Space Instead
1 
0 
0 
1 
0 
2
v1    , v2    , v3   
0 
 3
2
 
 
 
0 
 3
0 
1) Write down a matrix M whose row space is the span of
v1 , v2 and v3 .
M=
2) Using Wolfram-Alpha or Octave, row reduce M.
RREF(M) =
3) Are the vectors v1 , v2 and v3 linearly independent?
4) What is a basis for the span of v1 , v2 and v3 ?
Redoing 7B Using the Row Space Instead
1
2
  1
2
 3
 6
4
 5
,
,
,
v1    v 2    v3    v 4   
4
8
  6
6
 
 
 
 
  1
 2
0
 3
1) Write down a matrix M whose row space is the span of
v1 , v 2 , v 3 and v4 .
M=
2) Using clearly labeled row operations, row reduce M.
3) Are the vectors v1 , v 2 , v 3 and v4 linearly independent?
4) What is a basis for the span of v1 , v 2 , v 3 and v4 ?