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To determine if the vectors \((1, 0, 2)\), \((1, 0, 1)\), and \((0, 1, 2)\) are linearly independent or dependent, we can set up a linear independence test by forming a system of equations based on the following expression:

\[
c_1(1, 0, 2) + c_2(1, 0, 1) + c_3(0, 1, 2) = (0, 0, 0)
\]

This translates into the following equations:

1. \( c_1 + c_2 = 0 \)  (from the first component)
2. \( 0 \cdot c_1 + 0 \cdot c_2 + 1 \cdot c_3 = 0 \)  (from the second component)
3. \( 2c_1 + c_2 + 2c_3 = 0 \) (from the third component)

### Solving the System

1. From the first equation: \( c_1 = -c_2 \).
2. Substitute \( c_1 = -c_2 \) into the third equation to get:
   \[
   2(-c_2) + c_2 + 2c_3 = 0 \Rightarrow -2c_2 + c_2 + 2c_3 = 0 \Rightarrow -c_2 + 2c_3 = 0
   \]
3. Solve for \( c_2 \) in terms of \( c_3 \): \( c_2 = 2c_3 \).
4. Substituting \( c_2 = 2c_3 \) back into \( c_1 = -c_2 \), we get \( c_1 = -2c_3 \).

Since we have a nontrivial solution (the constants \( c_1 = -2c_3 \), \( c_2 = 2c_3 \), \( c_3 = c_3 \) can be non-zero for a specific \( c_3 \) value), the vectors are **linearly dependent**.

Would you like further details, or have any other questions?

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Here are some related questions to explore:

1. What is the geometric interpretation of linearly dependent vectors?
2. How can we test for linear independence using matrix determinants?
3. What happens if we add a fourth vector to this set? How does it affect dependence?
4. What are other methods for checking linear independence without solving equations?
5. How is linear independence useful in the context of vector spaces?

**Tip:** When determining linear independence of vectors, setting up a matrix with the vectors as columns and performing row reduction is often an efficient method.

Are the following vectors linearly independent or dependent? {(1, 0, 2), (1, 0, 1), (0, 1, 2)}

This problem tests whether the vectors (1, 0, 2), (1, 0, 1), and (0, 1, 2) are linearly independent or dependent. The solution involves setting up a system of equations and determining if a nontrivial solution exists, exploring the concept of linear dependence in vector spaces.

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