To determine the linear independence of the set of vectors $(1, 2, 3), (1, 1, 1), (0, 1, 2)$, we need to check if there exist scalars $c_1, c_2, c_3$ (not all zero) such that the following linear combination equals the zero vector:

$c_1(1, 2, 3) + c_2(1, 1, 1) + c_3(0, 1, 2) = (0, 0, 0)$

This can be written as the following system of linear equations by considering each component:

c_1(1) + c_2(1) + c_3(0) &= 0 \quad \text{(from the first components)} \\ c_1(2) + c_2(1) + c_3(1) &= 0 \quad \text{(from the second components)} \\ c_1(3) + c_2(1) + c_3(2) &= 0 \quad \text{(from the third components)} \end{aligned}$$ This simplifies to the following system of equations: $$\begin{aligned} c_1 + c_2 &= 0 \\ 2c_1 + c_2 + c_3 &= 0 \\ 3c_1 + c_2 + 2c_3 &= 0 \end{aligned}$$ ### Solving the system: 1. From the first equation, we have: $$c_2 = -c_1$$ 2. Substituting $$c_2 = -c_1$$ into the second equation: $$2c_1 + (-c_1) + c_3 = 0 \quad \Rightarrow \quad c_1 + c_3 = 0 \quad \Rightarrow \quad c_3 = -c_1$$ 3. Substituting $$c_2 = -c_1$$ and $$c_3 = -c_1$$ into the third equation: $$3c_1 + (-c_1) + 2(-c_1) = 0 \quad \Rightarrow \quad 3c_1 - c_1 - 2c_1 = 0 \quad \Rightarrow \quad 0 = 0$$ This last equation is trivially true, meaning it provides no new information. Thus, the system has free variables. ### Conclusion: The system has infinitely many solutions since all three variables $$c_1, c_2, c_3$$ depend on $$c_1$$. This means that the vectors are **linearly dependent**, because a non-trivial solution (with not all scalars zero) exists. Would you like further details on any part of this process? Here are five follow-up questions related to this topic: 1. How can we determine the rank of a set of vectors? 2. How do we test for linear dependence in larger sets of vectors? 3. What is the geometric interpretation of linear dependence in three-dimensional space? 4. Can a set of vectors be linearly dependent if there are more vectors than the dimension of the space? 5. How does the determinant of a matrix help in determining linear independence? **Tip:** A set of vectors in an $$n$$-dimensional space is linearly dependent if it contains more than $$n$$ vectors.