To show that the set $G = \{(a, 0) \mid a \in \mathbb{R}\}$ is a subspace of $\mathbb{R}^2$, we need to verify that it satisfies the three conditions for being a subspace:

1. The zero vector is in $G$.
2. $G$ is closed under vector addition.
3. $G$ is closed under scalar multiplication.

Let's go through these conditions one by one.

1. The Zero Vector is in $G$

The zero vector in $\mathbb{R}^2$ is $\mathbf{0} = (0, 0)$. For $(a, 0)$ to be in $G$, $a$ must be a real number. If we choose $a = 0$, then $(a, 0) = (0, 0) \in G$.

Thus, the zero vector is in $G$.

2. $G$ is Closed Under Vector Addition

Take any two vectors $\mathbf{u} = (a_1, 0)$ and $\mathbf{v} = (a_2, 0)$ from $G$, where $a_1, a_2 \in \mathbb{R}$.

The sum of these vectors is: $\mathbf{u} + \mathbf{v} = (a_1, 0) + (a_2, 0) = (a_1 + a_2, 0)$

Since $a_1 + a_2$ is a real number, $(a_1 + a_2, 0) \in G$.

Thus, $G$ is closed under vector addition.

3. $G$ is Closed Under Scalar Multiplication

Take any vector $\mathbf{u} = (a, 0) \in G$ and any scalar $c \in \mathbb{R}$.

The scalar multiple of $\mathbf{u}$ is: $c \mathbf{u} = c(a, 0) = (ca, c \cdot 0) = (ca, 0)$

Since $ca$ is a real number, $(ca, 0) \in G$.

Thus, $G$ is closed under scalar multiplication.

Conclusion

Since $G$ contains the zero vector, is closed under vector addition, and is closed under scalar multiplication, it satisfies all the conditions to be a subspace of $\mathbb{R}^2$. Therefore, $G = \{(a, 0) \mid a \in \mathbb{R}\}$ is indeed a subspace of $\mathbb{R}^2$.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What is the geometric interpretation of the subspace $G$ in $\mathbb{R}^2$?
2. How would you prove that a given set is not a subspace of $\mathbb{R}^2$?
3. Can a subspace of $\mathbb{R}^2$ be a one-dimensional line not passing through the origin?
4. What other subsets of $\mathbb{R}^2$ can be considered subspaces?
5. How does the definition of a subspace change when moving from $\mathbb{R}^2$ to $\mathbb{R}^3$?
6. If $G = \{(a, b) \mid a, b \in \mathbb{R}\}$, would $G$ still be a subspace of $\mathbb{R}^2$?
7. What are the properties of a subspace in terms of basis and dimension?
8. How can the concept of subspaces be extended to function spaces?

Tip: When verifying if a set is a subspace, always check for the presence of the zero vector, closure under addition, and closure under scalar multiplication.