The question asks whether the vector $\mathbf{v} = \begin{bmatrix} -1 \\ 4 \\ -1 \end{bmatrix}$ is in the null space (or kernel) of matrix $A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 1 & 1 \end{bmatrix}$.

To check if $\mathbf{v}$ is in the null space of $A$, we need to verify if the product of $A$ and $\mathbf{v}$ gives the zero vector:

$A \mathbf{v} = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} -1 \\ 4 \\ -1 \end{bmatrix}$

Let me compute this.The product of matrix $A$ and vector $\mathbf{v}$ is:

$A \mathbf{v} = \begin{bmatrix} 0 \\ 2 \end{bmatrix}$

Since the result is not the zero vector, the vector $\mathbf{v} = \begin{bmatrix} -1 \\ 4 \\ -1 \end{bmatrix}$ is not in the null space of matrix $A$.

Would you like further clarification or details on this?

Here are some related questions that could expand your understanding:

1. What does it mean for a vector to be in the null space of a matrix?
2. How can we find the null space of a matrix systematically?
3. What are the geometric interpretations of null space and its relationship to linear transformations?
4. How do we interpret the rank of a matrix in relation to its null space?
5. Can you explain the kernel and image of a matrix more deeply?

Tip: To check if a vector is in the null space of a matrix, the result of multiplying the matrix by the vector must yield the zero vector.