To determine all values of $x$ such that the set of vectors $\{(1, 1, 0), (-1, 0, -1), (0, 1, x)\}$ forms a basis for $\mathbb{R}^3$, we need to verify when these vectors span $\mathbb{R}^3$, meaning that they are linearly independent.

Three vectors in $\mathbb{R}^3$ span $\mathbb{R}^3$ if and only if they are linearly independent. The vectors are linearly independent if the determinant of the matrix formed by these vectors is non-zero.

Step 1: Construct the matrix

The matrix formed by placing the vectors as columns is:

1 & -1 & 0 \\ 1 & 0 & 1 \\ 0 & -1 & x \end{pmatrix}$$ ### Step 2: Compute the determinant We now compute the determinant of the matrix $$A$$. The determinant of a 3x3 matrix is given by: $$\text{det}(A) = 1 \cdot \det\begin{pmatrix} 0 & 1 \\ -1 & x \end{pmatrix} - (-1) \cdot \det\begin{pmatrix} 1 & 1 \\ 0 & x \end{pmatrix} + 0 \cdot \det\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$$ Simplifying each term: 1. $$\det\begin{pmatrix} 0 & 1 \\ -1 & x \end{pmatrix} = (0 \cdot x) - (-1 \cdot 1) = 1$$ 2. $$\det\begin{pmatrix} 1 & 1 \\ 0 & x \end{pmatrix} = (1 \cdot x) - (1 \cdot 0) = x$$ Thus, the determinant is: $$\text{det}(A) = 1 \cdot 1 + 1 \cdot x = 1 + x$$ ### Step 3: Set the determinant not equal to zero For the vectors to be linearly independent, the determinant must not be zero: $$1 + x \neq 0$$ Solving for $$x$$: $$x \neq -1$$ ### Conclusion The set of vectors $$\{(1, 1, 0), (-1, 0, -1), (0, 1, x)\}$$ spans $$\mathbb{R}^3$$ if and only if $$x \neq -1$$. --- Do you want further details or have any questions? Here are five related questions to deepen your understanding: 1. What is the geometric interpretation of linearly independent vectors in $$\mathbb{R}^3$$? 2. How can we check if vectors span a higher-dimensional space, such as $$\mathbb{R}^4$$? 3. What happens if the determinant of the matrix is zero? 4. Can a set of two vectors span $$\mathbb{R}^3$$? Why or why not? 5. How would the solution change if one of the vectors was the zero vector? **Tip:** The determinant of a matrix tells us if the transformation represented by that matrix has any "volume"; a zero determinant means that the space collapses into a lower dimension.