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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.

Determine whether w is in the column space of A, the null space of A, or both.

Explore a step-by-step solution to determine whether vector w is in the column space, null space, or both of matrix A. This problem uses concepts from linear algebra, including matrix multiplication and vector space properties, suitable for college-level students.

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