examin the linear independence of the following set of vector:- [(1,2,3),(1,1,1),(0,1,2)]

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:

\[
\begin{aligned}
    c_1(1) + c_2(1) + c_3(0) &= 0 \quad \text{(from the first components)} \\
    c_1(2) + c_2(1) + c_3(1) &= 0 \quad \text{(from the second components)} \\
    c_1(3) + c_2(1) + c_3(2) &= 0 \quad \text{(from the third components)}
\end{aligned}
\]

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.

Examine the linear independence of the following set of vectors: [(1,2,3),(1,1,1),(0,1,2)]

Analyze the linear independence of the set of vectors (1, 2, 3), (1, 1, 1), and (0, 1, 2). This problem explores key linear algebra concepts like linear independence and the solution of systems of equations. Detailed step-by-step explanation provided. Ideal for college-level students.

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