1. Show that set G = {(a, 0)/a ∈ R} is a subspace of R^2

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.

Learn how to prove that the set G = {(a, 0) | a ∈ ℝ} is a subspace of ℝ². This problem involves verifying closure under vector addition, scalar multiplication, and the presence of the zero vector. Suitable for advanced undergraduate students studying vector spaces and subspaces.

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