To solve this problem, we need to determine which of the vectors $w_1 = \begin{pmatrix} -1 \\ 2 \\ 5 \end{pmatrix}$ and $w_2 = \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$ can complete the set $\{ v_1, v_2 \}$ to form a basis for $\mathbb{R}^3$.

Given:

• $v_1 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$
• $v_2 = \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$

A basis for $\mathbb{R}^3$ requires three linearly independent vectors. Therefore, we need to check if adding either $w_1$ or $w_2$ to the set $\{ v_1, v_2 \}$ will result in a linearly independent set of three vectors.

Steps to Solve:

1. Linear Independence Check: Calculate the determinant of the matrix formed by $v_1$, $v_2$, and each $w_i$ (for $i = 1, 2$). If the determinant is non-zero, the vectors are linearly independent and form a basis for $\mathbb{R}^3$.

2. Set up Matrices:

   • For $\{ v_1, v_2, w_1 \}$: 1 & -1 & -1 \\ 2 & 0 & 2 \\ 3 & 1 & 5 \end{vmatrix}$$
   • For $\{ v_1, v_2, w_2 \}$: 1 & -1 & 0 \\ 2 & 0 & 1 \\ 3 & 1 & 1 \end{vmatrix}$$

3. Compute Determinants:

   • If the determinant of the first matrix is non-zero, $w_1$ completes the basis.
   • If the determinant of the second matrix is non-zero, $w_2$ completes the basis.

Let's compute these determinants:

1. Determinant for $\{ v_1, v_2, w_1 \}$: 1 & -1 & -1 \\ 2 & 0 & 2 \\ 3 & 1 & 5 \end{vmatrix} = 1 \cdot (0 \cdot 5 - 2 \cdot 1) - (-1) \cdot (2 \cdot 5 - 2 \cdot 3) + (-1) \cdot (2 \cdot 1 - 0 \cdot 3) = -2 + 4 + (-2) = 0$$ Since this determinant is zero, $$w_1$$ does not complete the basis.
2. Determinant for $\{ v_1, v_2, w_2 \}$: 1 & -1 & 0 \\ 2 & 0 & 1 \\ 3 & 1 & 1 \end{vmatrix} = 1 \cdot (0 \cdot 1 - 1 \cdot 1) - (-1) \cdot (2 \cdot 1 - 1 \cdot 3) + 0 = -1 + 1 \cdot (2 - 3) = -1 + (-1) = -2$$ Since this determinant is non-zero, $$w_2$$ does complete the basis.

Conclusion:

The answer is (b) only $w_2$.

Would you like a more detailed explanation or have further questions?

Here are 5 related questions to expand your understanding:

1. What are the properties of a basis in $\mathbb{R}^n$?
2. How can we determine linear independence using determinants?
3. What are the implications if the determinant of a set of vectors is zero?
4. How can row reduction help in determining linear independence?
5. Why does a basis in $\mathbb{R}^3$ require exactly three vectors?

Tip: For a quick check of linear independence, try using the determinant of a matrix formed by the vectors, as a non-zero value confirms independence.