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The question asks to find a basis for the subspace \( F \) of \( \mathbb{R}^3 \), where \( F \) is generated by the vectors \( (1, 2, -1), (2, 1, -3), (3, 0, -5) \). 

### Solution:
To find a basis for \( F \), we need to determine if the three vectors are linearly independent. If they are linearly independent, they form a basis for the subspace. If they are linearly dependent, we must eliminate the dependent vectors to get a basis.

We can check the linear independence by setting up a matrix with these vectors as rows or columns, then performing row reduction (Gaussian elimination).

The vectors given are:
\[
v_1 = (1, 2, -1), \quad v_2 = (2, 1, -3), \quad v_3 = (3, 0, -5)
\]

Form the matrix with these vectors as rows:
\[
A = \begin{pmatrix}
1 & 2 & -1 \\
2 & 1 & -3 \\
3 & 0 & -5
\end{pmatrix}
\]

Next, perform Gaussian elimination on this matrix.

1. Subtract 2 times row 1 from row 2:
   \[
   R_2 \rightarrow R_2 - 2R_1
   \]
   This gives:
   \[
   \begin{pmatrix}
   1 & 2 & -1 \\
   0 & -3 & -1 \\
   3 & 0 & -5
   \end{pmatrix}
   \]

2. Subtract 3 times row 1 from row 3:
   \[
   R_3 \rightarrow R_3 - 3R_1
   \]
   This gives:
   \[
   \begin{pmatrix}
   1 & 2 & -1 \\
   0 & -3 & -1 \\
   0 & -6 & -2
   \end{pmatrix}
   \]

3. Now, divide row 2 by \(-3\):
   \[
   R_2 \rightarrow \frac{1}{-3} R_2
   \]
   This gives:
   \[
   \begin{pmatrix}
   1 & 2 & -1 \\
   0 & 1 & \frac{1}{3} \\
   0 & -6 & -2
   \end{pmatrix}
   \]

4. Add 6 times row 2 to row 3:
   \[
   R_3 \rightarrow R_3 + 6R_2
   \]
   This gives:
   \[
   \begin{pmatrix}
   1 & 2 & -1 \\
   0 & 1 & \frac{1}{3} \\
   0 & 0 & 0
   \end{pmatrix}
   \]

The third row is now all zeros, meaning that the third vector is a linear combination of the first two. Therefore, the vectors \( v_1 \) and \( v_2 \) form a basis for \( F \).

### Answer:
The basis for \( F \) is:
\[
\{ (1, 2, -1), (2, 1, -3) \}
\]

Thus, the correct option is **b.** \( \{(1, 2, -1), (2, 1, 0)\} \), but the correct vectors should actually be \( (1, 2, -1) \) and \( (2, 1, -3) \). The mistake might be in the problem's options.

Would you like more details or have any other questions?

### Related Questions:
1. What is the definition of a basis in linear algebra?
2. How can we determine if vectors are linearly dependent or independent?
3. What is Gaussian elimination, and how does it help in finding bases?
4. Why does having a row of zeros in the row-reduced form imply linear dependence?
5. How do the dimensions of a subspace relate to the number of vectors in its basis?

### Tip:
Always check the linear independence of vectors using Gaussian elimination to confirm whether they form a basis!

Find a basis for the subspace F of R^3, where F is generated by the vectors (1, 2, -1), (2, 1, -3), (3, 0, -5).

Learn how to find a basis for the subspace F in R^3 generated by the vectors (1, 2, -1), (2, 1, -3), and (3, 0, -5) using linear independence and Gaussian elimination. Understand the process of row reduction to identify a linearly independent subset. Ideal for undergraduate linear algebra students.

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