Say whether the following sets are vector subspaces of the vector space V in each of the following cases: 1) S1 = {(x, y) ∈ R 2 : |x| = |y|} , V = R 2

To determine whether the set $S_1 = \{(x, y) \in \mathbb{R}^2 : |x| = |y|\}$ is a vector subspace of the vector space $V = \mathbb{R}^2$, we will check two main criteria for a subset to be a subspace:

1. Non-emptiness/Contains the Zero Vector
2. Closure under Addition and Scalar Multiplication

Step 1: Non-emptiness/Contains the Zero Vector

We need to determine if the zero vector is in $S_1$.

• The zero vector in $\mathbb{R}^2$ is $(0, 0)$.
• We check if $|0| = |0|$.
• This is true, as $0 = 0$.

Thus, $(0, 0) \in S_1$.

Step 2: Closure under Addition

We need to check if for any two vectors $(x_1, y_1)$ and $(x_2, y_2)$ in $S_1$, the vector $(x_1 + x_2, y_1 + y_2)$ is also in $S_1$.

Assume:

• $|x_1| = |y_1|$
• $|x_2| = |y_2|$

We need to check if: $|(x_1 + x_2)| = |(y_1 + y_2)|$

This equality does not generally hold. For example, if we take $(1, 1)$ and $(-1, -1)$, both of which are in $S_1$:

• Their sum is $(1 + (-1), 1 + (-1)) = (0, 0)$, which is okay.
• If we take $(1, 1)$ and $(1, -1)$ instead, $|(1+1)| = 2$ and $|(1 + (-1))| = 0$, which means $|x| \neq |y|$.

Step 3: Closure under Scalar Multiplication

Now we need to check if for any vector $(x, y) \in S_1$ and any scalar $c \in \mathbb{R}$, the vector $(cx, cy)$ is also in $S_1$:

Assume $|x| = |y|$:

• Check $|cx| = |cy|$

As $|cx| = |c||x|$ and $|cy| = |c||y|$, it holds if $|c||x| = |c||y|$, which is true if $c \neq 0$. However, for $c = 0$, it leads to $(0, 0)$, which we already established is in $S_1$.

Step 4: Conclusion

Since the set is not closed under vector addition (as shown in Step 2), we conclude:

Final Answer

The set $S_1$ is not a vector subspace of $V = \mathbb{R}^2$.