Let $H=\left\{\left[\begin{array}{l}x \\ y\end{array}\right]: 5 x^{2}+y^{2} \leq 1\right\}$, which represents the set points on and inside an ellipse in the $x y$-plane. Find two specific examples - two vectors, and a vector and a scalar-to show that $\mathrm{H}$ is not a subspace of $\mathbb{R}^{2}$

Step 1 ：Let $H=\left\{\left[\begin{array}{l}x \\ y\end{array}\right]: 5 x^{2}+y^{2} \leq 1\right\}$, which represents the set points on and inside an ellipse in the $x y$-plane.

Step 2 ：To show that a set H is not a subspace of R^2, we need to show that it does not satisfy one of the three properties of a subspace: 1. The zero vector is in H. 2. For every vector in H, its negative is also in H. 3. For every vector in H and every scalar, the scalar multiplication of the vector is also in H.

Step 3 ：In this case, we can see that the set H does not satisfy the third property. For example, if we take the vector [0.5, 0] which is in H, and multiply it by the scalar 2, we get the vector [1, 0] which is not in H because 5*(1)^2 + 0^2 = 5 > 1.

Step 4 ：Final Answer: The set $H=\left\{\left[\begin{array}{l}x \\ y\end{array}\right]: 5 x^{2}+y^{2} \leq 1\right\}$ is not a subspace of $\mathbb{R}^{2}$ because it does not satisfy the third property of a subspace. Two specific examples that demonstrate this are the vector [0.5, 0] and the scalar 2. When we multiply the vector by the scalar, we get a vector [1, 0] that is not in H. Therefore, $\boxed{H \text{ is not a subspace of } \mathbb{R}^{2}}$.