Example 1: Testing for Linear Independence

Problem: Is the following set of vectors linearly dependent?

Where:

For a set of vectors to be linearly independent, the only solution to the equation:

must be and

In this case:

If not only the zero solution exists (i.e., if or can be non-zero), the set is linearly dependent.

Step 1. Set up the system of equations:

2. Eliminate one variable

Now the system is

3. Subtract the equations

Simplifies to:

So:

4. Substitute back to find

Now that we know , substitute this value into one of the original equations. Let’s use the second equation:

Substitute :

Conclusion:

Since , the set of vectors is linearly independent. These vecots span .

Example 2: Testing for Linear Dependence

Problem: Is the following set of vectors linearly dependent?

Where:

The span of this set is the collection of all vectors that can be formed by linear combinations of and :

Since , the linear combination becomes:

Thus, any linear combination of these vectors is just a scalar multiple of . The span is a single line in , and the vectors are linearly dependent.

For any two colinear vectors in , their span reduces to a single line.

One vector in the set can be represented by some combination of other vectors in the set