Linear Dependence or Independence
The set of vectors (1,2,1), (0,1,0), (3,4,3) in R^3 can be analyzed for linear dependence or independence.

Definition of Linear Dependence and Independence
- A set of vectors is linearly dependent if there exist scalars, not all zero, such that a linear combination of the vectors equals the zero vector.
- A set of vectors is linearly independent if the only way to obtain the zero vector as a linear combination is by setting all scalars to zero.

Analysis of the Given Set
- Let's consider the vectors as columns of a matrix and row-reduce it to determine linear dependence.
- Construct the matrix with the given vectors as columns:
\[ \begin{bmatrix} 1 & 0 & 3 \\ 2 & 1 & 4 \\ 1 & 0 & 3 \end{bmatrix} \]
- Perform row operations to row-reduce the matrix.
- After row-reducing, if we have a row of zeros, then the set is linearly dependent.

Evaluation of the Set
- Upon row-reducing the matrix, if we find a row of zeros, then the set is linearly dependent.
- If no row of zeros is obtained, then the set is linearly independent.

Conclusion
- Check the row-reduced matrix to determine if a row of zeros is present.
- Based on the presence or absence of a row of zeros, conclude whether the set is linearly dependent or independent.
- This determination will help decide if the given set is a basis for R^3 or not.