1 point) Let H be the set of all points in the second quadrant in the plane V =

Question

1 point) Let H be the set of all points in the second quadrant in the plane V = R2. That is, H = {(r, y) 1 x subspace of the vector space V? 0, y 0). Is H a 1. Is H nonempty? choose 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 syntax such as , . 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, using a comma separated list and syntax such as 2, . 4. Is H a subspace of the vector space V? You should be able to justify your answer using the subspace test (Theorem 4.5) and your answers to parts 1-3 choose

Explanation / Answer

The set H is non-empty as the point (-2,3) ( =an illustration only) is in H. If X1 = (x1,y1) and X2 = (x2,y2) are 2 arbitrary points in H, then x1 0, x2 0,y1 0 and y2 0. Further, X1+X2 = (x1,y1) + (x2,y2)= (x1+x2,y1+y2). Now, x1+x2 0 and y1+y2 0 so that X1+X2 is in H. Hence H is closed under vector addition. Let X1 = (x1,y1) be an arbitrary point in H and let be an arbitrary scalar. Then x1 0, and y1 0. Further, X = (x1,y1)= (x1,y1). Now, when is a negative scalar, x1 0 and y1 0. Thus, X is not in H when is negative. Hence H is not closed under scalar multiplication. H is not a vector space as it is not closed under scalar multiplication. Hence H is not a subspace of V = R2.

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