To determine whether vector $w$ is in the column space of $A$, the null space of $A$, or both, let's go through these two checks step by step:

1. Checking if $w$ is in the Column Space of $A$:

To determine if $w$ is in the column space of $A$, we need to see if there exists a vector $x$ such that $A \cdot x = w$. In other words, we need to solve the linear system: $A \cdot x = w$ where $w = \begin{bmatrix} 1 \\ 2 \\ 1 \\ 0 \end{bmatrix} \quad \text{and} \quad A = \begin{bmatrix} -11 & 7 & -3 & 0 \\ -5 & 2 & 1 & -4 \\ 13 & -10 & 7 & -6 \\ 3 & -2 & 1 & 0 \end{bmatrix}.$ If this system has a solution, then $w$ is in the column space of $A$; otherwise, it is not.

2. Checking if $w$ is in the Null Space of $A$:

To determine if $w$ is in the null space of $A$, we need to check if $A \cdot w = 0$. This requires calculating the product: $A \cdot w = \begin{bmatrix} -11 & 7 & -3 & 0 \\ -5 & 2 & 1 & -4 \\ 13 & -10 & 7 & -6 \\ 3 & -2 & 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 1 \\ 0 \end{bmatrix}.$ If the result of this multiplication is the zero vector, then $w$ is in the null space of $A$; otherwise, it is not.

I'll proceed with these calculations to confirm if $w$ lies in the column space, the null space, or both.The results are as follows:

1. Null Space Check: The product $A \cdot w = 0$, which means $w$ is in the null space of $A$.
2. Column Space Check: There exists a solution to $A \cdot x = w$, which confirms that $w$ is also in the column space of $A$.

Conclusion

Vector $w$ is in both the column space and the null space of $A$.

Would you like further details on any part of this solution?

Here are some additional questions to deepen understanding:

1. How do we determine if a vector lies in the row space of a matrix?
2. What does it mean for a vector to be in both the column space and the null space of a matrix?
3. How can we interpret the solution $x$ in the context of this problem?
4. What properties of $A$ allow $w$ to be in both spaces?
5. How would the solution change if $w$ were a different vector?

Tip: For a vector to be in the null space of a matrix, multiplying the matrix by the vector should yield the zero vector.