To determine if the set of vectors $\{(1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 0, 1)\}$ is linearly independent or dependent, we need to check if there exists a non-trivial solution to the equation: $c_1(1, 0, 0) + c_2(1, 1, 0) + c_3(1, 1, 1) + c_4(1, 0, 1) = (0, 0, 0)$ where $c_1$, $c_2$, $c_3$, and $c_4$ are scalars.

This translates to a system of equations based on the components:

1. $c_1 + c_2 + c_3 + c_4 = 0$
2. $c_2 + c_3 = 0$
3. $c_3 + c_4 = 0$

Let's analyze this system to find if there are non-zero solutions for $c_1$, $c_2$, $c_3$, and $c_4$. If such solutions exist, the vectors are linearly dependent; otherwise, they are linearly independent.

Solving this system, we see that it is possible to find non-zero values for $c_1$, $c_2$, $c_3$, and $c_4$ that satisfy these equations, meaning the vectors are linearly dependent.

Would you like a detailed step-by-step solution on how this was determined?

1. What is the definition of linear dependence and independence in the context of vector spaces?
2. How can we determine if a larger set of vectors is linearly dependent or independent?
3. What are the implications of linear dependence for vector spaces and basis selection?
4. How can matrix row reduction help in determining linear dependence?
5. What is the role of the rank of a matrix in checking linear dependence?

Tip: To test for linear dependence, setting up the system of equations based on vector components is essential for simplifying the process.