1. Linear Independence Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. Linear Independence
A set of vectors in ℝm is said to be linearly independent if the vector equation
has only the trivial solution x1=x2=…=xn=0
If there is some nonzero solution then the set of vectors is linearly dependent.
This implies some redundancy in the set of vectors.
i.e. we can write one of the vectors in terms of the others.
When we put the vectors in a matrix and do row reduction, the number of pivots (or basic variables) corresponds to the dimension of the span of the set.
If there are any free variables, then the set is dependent.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. Linear Independence
Example: Are the following sets of vectors linearly independent?
Describe the span of each set.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4. Linear Independence
Example: Are the following sets of vectors linearly independent?
Describe the span of each set.
V1 spans ℝ2.
We will test this one with the definition.
Try to solve this equation:
This is equivalent to a system of 2 equations:
Or an augmented matrix, with the vectors as columns:
There is no nonzero solution, so the set is independent.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. Linear Independence
Example: Are the following sets of vectors linearly independent?
Describe the span of each set.
V2 spans ℝ2.
Notice that this set contains 3 vectors from ℝ2, and we already know that the first 2 vectors are independent (from the previous set).
V2 must be a dependent set. The dimension of the vectors determines a maximum number of independent vectors that can be in a set. The span of this set is ℝ2. You could say that the third vector doesn’t add anything new– it must be in the span of the others.
Another way to see that the set is dependent is to write one of the vectors in terms of the other vectors in the set. This amounts to rearranging the equation
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. Linear Independence
Example: Are the following sets of vectors linearly independent?
Describe the span of each set.
V3 spans ℝ3.
Set up the augmented matrix for this one. It is already in RREF.
Since there is only the zero solution, the set is independent.
Note: This set is called the STANDARD BASIS for ℝ3.
In general, for ℝn the standard basis will have n vectors, each with a single 1 and zeroes elsewhere, so that they form the nxn identity matrix.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. Linear Independence
Example: Are the following sets of vectors linearly independent?
Describe the span of each set.
V4 spans a 2-dimensional subset of ℝ3 (a plane).
Let’s reduce the augmented matrix for this one.
The RREF matrix has a non-pivot column, which means there is a free variable. There are 2 pivot positions, so the span of this set must be 2-dimensional (a plane in ℝ3). This set is linearly dependent.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8. Linear Independence
Example: Are the following sets of vectors linearly independent?
Describe the span of each set.
V4 spans a 2-dimensional subset of ℝ3 (a plane).
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB