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To determine if vector **w** is in the null space of matrix \( A \), we need to check if \( A \cdot \mathbf{w} = \mathbf{0} \). In other words, we multiply matrix \( A \) by vector \( \mathbf{w} \) and see if the result is the zero vector.

### Given:
\[
\mathbf{w} = \begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}
\]
\[
A = \begin{bmatrix} 4 & -1 & -3 \\ 6 & -1 & 4 \\ -2 & 3 & 2 \end{bmatrix}
\]

### Step 1: Compute \( A \cdot \mathbf{w} \)

To find \( A \cdot \mathbf{w} \), we perform the matrix-vector multiplication:

\[
A \cdot \mathbf{w} = \begin{bmatrix} 4 & -1 & -3 \\ 6 & -1 & 4 \\ -2 & 3 & 2 \end{bmatrix} \cdot \begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}
\]

#### Calculations:
1. First row: \( 4(1) + (-1)(5) + (-3)(-1) = 4 - 5 + 3 = 2 \)
2. Second row: \( 6(1) + (-1)(5) + 4(-1) = 6 - 5 - 4 = -3 \)
3. Third row: \( -2(1) + 3(5) + 2(-1) = -2 + 15 - 2 = 11 \)

So,
\[
A \cdot \mathbf{w} = \begin{bmatrix} 2 \\ -3 \\ 11 \end{bmatrix}
\]

### Conclusion
Since \( A \cdot \mathbf{w} \neq \mathbf{0} \), vector \( \mathbf{w} \) is **not** in the null space of \( A \).

Would you like further details on any of these steps, or do you have any questions?

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Here are some related questions to consider:
1. What is the geometric interpretation of the null space of a matrix?
2. How can we determine if a vector is in the column space of a matrix?
3. What methods can be used to compute the null space of a matrix?
4. How does row reduction help in finding solutions to \( A \cdot \mathbf{x} = 0 \)?
5. What is the rank-nullity theorem and how does it relate to this problem?

**Tip:** For any matrix \( A \), the null space represents all solutions to the equation \( A \cdot \mathbf{x} = 0 \).

Determine if w = [1, 5, -1] is in Nul A, where A = [[4, -1, -3], [6, -1, 4], [-2, 3, 2]].

Find out if vector w = [1, 5, -1] is in the null space of matrix A = [[4, -1, -3], [6, -1, 4], [-2, 3, 2]] by performing matrix-vector multiplication. This problem involves core concepts from linear algebra and is suitable for college-level students.

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