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?

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 $A \cdot \mathbf{x} = 0$.