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The problem you provided asks about the linear independence of two sets of vectors formed from three given linearly independent vectors \( x, y, \) and \( z \) in \( \mathbb{R}^n \).

### Part (a)
**Question**: Are the vectors \( (x - y), (y - z), (z - x) \) linearly independent?

To determine the linear independence of \( (x - y), (y - z), (z - x) \), we check if the only solution to the equation
\[ a(x-y) + b(y-z) + c(z-x) = 0 \]
is \( a = b = c = 0 \).

Expanding and rearranging the terms, we get:
\[ a x - a y + b y - b z + c z - c x = 0 \]
\[ (a - c)x + (b - a)y + (c - b)z = 0 \]

Since \( x, y, \) and \( z \) are linearly independent, the coefficients of \( x, y, \) and \( z \) must each be zero:
\[ a - c = 0 \]
\[ b - a = 0 \]
\[ c - b = 0 \]

These equations imply \( a = b = c \). This means the equation can be true for any non-zero scalar \( a = b = c \neq 0 \). Hence, the vectors \( (x-y), (y-z), (z-x) \) are **not linearly independent** as they are linearly dependent on each other.

### Part (b)
**Question**: Are the vectors \( (x + y), (y + z), (z + x) \) linearly independent?

To determine the linear independence of \( (x + y), (y + z), (z + x) \), we consider the equation:
\[ a(x+y) + b(y+z) + c(z+x) = 0 \]

Expanding this, we get:
\[ a x + a y + b y + b z + c z + c x = 0 \]
\[ (a + c)x + (a + b)y + (b + c)z = 0 \]

Again, using the linear independence of \( x, y, \) and \( z \), we must have:
\[ a + c = 0 \]
\[ a + b = 0 \]
\[ b + c = 0 \]

These equations imply \( a = -b, b = -c, \) and \( c = -a \). Solving these gives \( a = b = c = 0 \). Hence, the vectors \( (x + y), (y + z), (z + x) \) are **linearly independent**.

Would you like further details on any part or have any other questions about linear independence or vector spaces? Here are some related questions and a tip:

1. How can you determine linear independence in higher dimensions?
2. What other types of vector operations might affect linear independence?
3. Can adding more vectors to a set ever lead to a loss of linear independence?
4. What are the implications of linear independence in solving systems of linear equations?
5. How do properties of linear independence apply to function spaces?

**Tip**: When solving problems involving vector independence, organizing terms by vector components can simplify the visualization of the problem and solutions.

Let x, y, and z be 3 linearly independent vectors in R^n. Are the vectors (x-y), (y-z), (z-x) linearly independent? Are the vectors (x+y), (y+z), (z+x) linearly independent?

Explore how to determine the linear independence of vector combinations using three linearly independent vectors x, y, and z in R^n. This solution examines two different sets of vector combinations: (x-y), (y-z), (z-x) and (x+y), (y+z), (z+x), providing step-by-step explanations and insights on linear independence and vector spaces.

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