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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.

If A = [[1, 0, -1], [1, 1, 1]], is v = [-1, 4, -1] in null(A)?

Explore whether the vector [-1, 4, -1] is in the null space of matrix A = [[1, 0, -1], [1, 1, 1]]. This article provides a detailed step-by-step solution using matrix multiplication to determine membership in the null space.

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