Final answer:
To determine if vectors are linearly independent, one must find a dependence relation by setting up a matrix with the vectors, performing row reduction to RREF, and analyzing the result. A non-trivial solution to this system indicates linear dependence, while the trivial solution (all zeros) indicates linear independence.
Step-by-step explanation:
The question provided relates to the concept of linear independence of vectors in mathematics, specifically in the context of linear algebra. To determine if a set of vectors are linearly independent, we look for a dependence relation. If no such relation exists, then the vectors are indeed linearly independent. A dependence relation would be coefficients that are not all zero, which when multiplied by the respective vectors, would result in the zero vector. If such coefficients can be found, the vectors are said to be linearly dependent.
To solve this, you would typically set up an equation where the vectors are multiplied by scalars and set equal to the zero vector. Then, solve the system of equations to determine the scalars. The presence of non-trivial solutions (scalars that are not all zero) indicates linear dependence. If only the trivial solution exists (all scalars equal to zero), then the vectors are linearly independent.
Here are the steps commonly used to solve such problems:
- Set up the matrix with the vectors as columns.
- Perform row reduction to obtain the reduced row echelon form (RREF) of the matrix.
- Analyze the RREF to determine if there is a row of zeros, which would indicate that the vectors are linearly dependent.
If a dependence relation is found, we express it in terms of the vectors and the coefficients found. If not, we conclude the vectors are linearly independent.