Question

Let H be the set of all points in the first quadrant in the plane V=R 2 . That is, H={(x,y)∣x≥0,y≥0}. Is H a subspace of the vector space V ? 1. Is H nonempty? 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and svntax such as ⟨1,2⟩,⟨3,4⟩. 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a vector in H whose product is not in H; IIsinn a comma separated list and syntax such as 2,⟨3,4⟩ 4. Is H a subspace of the vector space V ? You should be able to justify your answer by writing a complete, coherent, and detailed proof based nn vour answers to parts 1-3.

Answer 1

H is a subspace of vector space V because it is nonempty, closed under addition, and closed under scalar multiplication.

Step-by-step explanation:

In order to determine if H is a subspace of vector space V, we need to consider three conditions:

1. H must be nonempty, meaning it must contain at least one vector. In this case, H is nonempty because it contains the point (0,0).
2. H must be closed under addition. To test this, we can take two vectors in H, (a,b) and (c,d), and add them together to see if the result, (a+c,b+d), is also in H. Since H is defined as the set of all points in the first quadrant, the sum of two vectors in H will also be in the first quadrant, so H is closed under addition.
3. H must be closed under scalar multiplication. To test this, we can multiply a vector in H, (a,b), by a scalar, k, and see if the result, (ka,kb), is also in H. Again, since H is defined as the set of all points in the first quadrant, any scalar multiple of a vector in H will also be in the first quadrant, so H is closed under scalar multiplication.

Since H satisfies all three conditions, it is a subspace of vector space V.

Answer 2

H is not a subspace of the vector space V because it is not closed under vector addition and scalar multiplication, failing two of the three required conditions for being a subspace.

Step-by-step explanation:

When considering whether a set H is a subspace of a vector space V, three conditions need to be satisfied: H must be nonempty, closed under vector addition, and closed under scalar multiplication. In this case, H is the set of all points in the first quadrant of the plane (two-dimensional vectors with non-negative components).

1. Is H nonempty

H contains the zero vector <0,0>, so it is nonempty.

2. Is H closed under addition

H is not closed under addition because if we take two vectors <0,1> and <1,0> that both belong to H, their sum <1,1> is in H but taking vectors with negative components, like <-2,2> and <2,-2>, while each vector individually lies in the respective quadrants, their sum <0,0> is in H, a system like this is not closed under addition.

3. Is H closed under scalar multiplication?

H is not closed under scalar multiplication because if we take a negative scalar, e.g., (-1), and multiply it by any vector in H, the resultant vector will have negative components and thus not lie in H. An example would be: -1 * <1,1> = <-1,-1>, which is not in H.

4. Is H a subspace of the vector space V?

Since H fails to be closed under vector addition and scalar multiplication, H is not a subspace of V.