We define linear independence of a set of vectors, and explore this concept algebraically and geometrically.
VEC-0100: Linear Independence
Consider the vector equation
where are vectors of and are unknown scalar coefficients.
Can we say whether or not this equation is consistent?
Observe that this equation must be consistent, regardless of what the vectors are, because is a solution. This solution is known as the trivial solution.
(Click the arrow on the right for the answer.)
Linear Independence Let be vectors of . We say that the set is linearly independent
if the only solution to
is the trivial solution .
If, in addition to the trivial solution, a non-trivial solution (not all are zero) exists, then we say that the set is linearly dependent.
If the set of vectors is linearly independent (dependent), we often say that the vectors are linearly independent (dependent).
Determine whether the vectors in each part are linearly independent.
item:linindpart1 We will solve the vector equation
Clearly is a solution to the equation. The question is whether another solution exists.
The vector equation translates into the following system:
Writing the system in augmented matrix form and applying elementary row operations gives us the following reduced row-echelon form: This shows that (eq:linrelationpart1) has infinitely many solutions: Letting , we obtain the following:
We conclude that the vectors are linearly dependent.
item:linindpart2 We need to solve the equation
Converting the equation to augmented matrix form and performing row reduction gives us This shows that is the only solution. Therefore the two vectors are linearly independent.
In Part item:linindpart1 of Example ex:linind we found that the vectors are linearly dependent because
Observe that (eq:nontrivrel) allows us to solve for one of the vectors and express it as a linear combination of the others. For example, This would not be possible if a nontrivial solution to the equation did not exist.
A subset of containing two or more vectors is linearly dependent if and
only if one of the vectors can be expressed as a linear combination of the
others.
- Proof
- See Practice Problem prob:lindeplincombofother.
Geometry of Linearly Dependent and Linearly Independent Sets
Theorem th:lindeplincombofother gives us a convenient way of looking at linear dependence/independence geometrically. When looking at two or more vectors, we ask “can one of the vectors be written as a linear combination of the others?” If the answer is “yes”, then the vectors are linearly dependent.
A Set of Two Vectors
Two vectors are linearly dependent if and only if one is a scalar multiple of the other. Two nonzero linearly dependent vectors may look like this:
or like this:
Two linearly independent vectors will look like this:
A Set of Three Vectors
Given a set of three nonzero vectors, we have the following possibilities:
- (Linearly Dependent Vectors) The three vectors are scalar multiples of each
other.
- (Linearly Dependent Vectors) Two of the vectors are scalar multiples of each
other.
- (Linearly Dependent Vectors) One vector can be viewed as the diagonal of a
parallelogram determined by scalar multiples of the other two vectors. All three
vectors lie in the same plane.
- (Linearly Independent Vectors) A set of three vectors is linearly independent if the vectors do not lie in the same plane. For example, vectors , and are linearly independent.
Practice Problems
Each problem below provides information about vectors . If possible, determine
whether the vectors are linearly dependent or independent.
The vectors are linearly independent The vectors are linearly dependent There is not enough information given to make a determination
Prove Theorem th:lindeplincombofother.
This is an “if and only if” statement. First, assume that the set is
linearly dependent and show that one vector can be expressed as a linear
combination of the others. Second, assume that one vector can be expressed
as a linear combination of the others, then shown that the set is linearly
dependent.
Each diagram below shows a collection of vectors. Are the vectors linearly dependent
or independent?
The vectors are linearly independent The vectors are linearly dependent There is not enough information given to make a determination
The vectors are linearly independent The vectors are linearly dependent There is not enough information given to make a determination
