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.

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 $\mathbb{R}^n$.

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.

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.