Let's analyze the question.

Given:

• The set $W$ is defined as the union of the first and third quadrants in the $xy$-plane, i.e., $W = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} : xy \geq 0 \right\}$.
• $u = \begin{pmatrix} x \\ y \end{pmatrix}$ is a vector in $W$.
• We need to determine if $c \cdot u$, where $c$ is any scalar, is also in $W$, and we need to choose the correct option from A, B, or C.

Analysis:

• For $u$ to be in $W$, the product $xy$ must be non-negative ($xy \geq 0$).
• If $c$ is a scalar, then $c \cdot u = \begin{pmatrix} cx \\ cy \end{pmatrix}$.

Consider the cases for $c$:

1. If $c > 0$:

   • The signs of $x$ and $y$ remain the same, so $(cx)(cy) = c^2(xy)$. Since $c^2 > 0$, $(cx)(cy)$ will still be non-negative. Hence, $c \cdot u$ is in $W$.

2. If $c = 0$:

   • $c \cdot u = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$, which is the zero vector. The zero vector is trivially in $W$ because $0 \times 0 = 0 \geq 0$.

3. If $c < 0$:

   • The signs of $cx$ and $cy$ are opposite to those of $x$ and $y$, but $(cx)(cy) = c^2(xy)$. Since $c^2 > 0$, the product is still non-negative. Thus, $c \cdot u$ is still in $W$.

Conclusion:

Regardless of whether $c$ is positive, negative, or zero, the product $(cx)(cy) = c^2(xy)$ remains non-negative, which means $c \cdot u$ is still in $W$.

Correct Answer: Option A is correct because it correctly explains that $c \cdot u$ will always be in $W$ because $c^2(xy) \geq 0$ whenever $xy \geq 0$.

Do you want more details or have any questions?

Related Questions:

1. What would change if $W$ were defined differently, such as only including one quadrant?
2. How does the union of the first and third quadrants affect the properties of $W$?
3. What does it mean for a set to be closed under scalar multiplication?
4. Can you provide an example of a set that is not closed under scalar multiplication?
5. How would the answer change if the condition $xy > 0$ were used instead?

Tip:

When dealing with vector spaces or sets defined by inequalities, always check the conditions for each quadrant and the effect of scalar multiplication on those conditions.