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Let's solve the problem step by step.

We are tasked with proving that \( N(A) = \{x \mid Ax = 0\} \), which is the null space of the matrix \(A\), is a subspace. We will use the three properties that define a subspace in vector spaces:

### Requirements for Subspace:
To prove that \( N(A) \) is a subspace of \( \mathbb{R}^n \), we need to verify these three properties:

1. **The zero vector is in \(N(A)\).**
2. **Closure under addition:** If \( x_1 \) and \( x_2 \) are in \(N(A)\), then \( x_1 + x_2 \) is also in \(N(A)\).
3. **Closure under scalar multiplication:** If \( x \) is in \(N(A)\) and \( c \) is a scalar, then \( cx \) is also in \(N(A)\).

---

### Step-by-step Verification:

#### 1. The zero vector is in \(N(A)\):
We need to check if the zero vector \( \mathbf{0} \) is in \(N(A)\). 

By definition, \( \mathbf{0} \in N(A) \) if:
\[
A\mathbf{0} = 0
\]
Since multiplying any matrix \( A \) by the zero vector gives the zero vector, we have:
\[
A\mathbf{0} = 0
\]
Thus, the zero vector is indeed in \(N(A)\).

#### 2. Closure under addition:
Let \( x_1, x_2 \in N(A) \), which means:
\[
Ax_1 = 0 \quad \text{and} \quad Ax_2 = 0
\]
We need to show that \( x_1 + x_2 \in N(A) \), i.e., that \( A(x_1 + x_2) = 0 \).

Using the linearity of matrix multiplication, we have:
\[
A(x_1 + x_2) = Ax_1 + Ax_2 = 0 + 0 = 0
\]
Thus, \( x_1 + x_2 \in N(A) \), proving that \(N(A)\) is closed under addition.

#### 3. Closure under scalar multiplication:
Let \( x \in N(A) \) and \( c \) be a scalar. We need to show that \( cx \in N(A) \), i.e., that \( A(cx) = 0 \).

Again, using the linearity of matrix multiplication, we have:
\[
A(cx) = c(Ax) = c \cdot 0 = 0
\]
Thus, \( cx \in N(A) \), proving that \(N(A)\) is closed under scalar multiplication.

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### Subspace of \( \mathbb{R}^m \) or \( \mathbb{R}^n \)?
If \( A \) is an \( m \times n \) matrix, then \( x \) (the elements of \( N(A) \)) are vectors from \( \mathbb{R}^n \), since \( A \) acts on vectors from \( \mathbb{R}^n \). Thus, the null space \( N(A) \) is a subspace of \( \mathbb{R}^n \).

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### Conclusion:
We have verified that \( N(A) \) satisfies all three properties for being a subspace. Therefore, \( N(A) \) is a subspace of \( \mathbb{R}^n \) if \( A \) is an \( m \times n \) matrix.

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Would you like any more details on any of these steps or further clarifications?

Here are 5 follow-up questions to deepen understanding:

1. How would you prove that the column space of \( A \) is a subspace?
2. What is the relationship between the null space and the rank of a matrix?
3. How does the dimension of \( N(A) \) relate to the rank-nullity theorem?
4. Can the null space of a matrix ever be all of \( \mathbb{R}^n \)? If so, when?
5. What happens to the null space if matrix \( A \) is invertible?

**Tip:** Always remember that any subspace must include the zero vector, and checking this first is a good starting point in proving subspace properties.

Show formally, by going through the three requirements for subspaces, that N(A) = {x | Ax = 0} is a subspace. Do each part separately. If A is a m x n matrix, is N(A) a subspace of R^m or R^n?

A step-by-step guide to proving that the null space N(A) = {x | Ax = 0} is a subspace by using the three properties required for subspaces: closure under addition, scalar multiplication, and the inclusion of the zero vector. This problem explores concepts in linear algebra, specifically related to subspaces and matrices.

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