How to prove a subset of a vector space is a subspace?
Answers
ifw1+w2=(a1+a2,b1+b2,c1+c2)then it satisfies that the first coordinate is greater than or equal to the second coordinate. Is a1+a2≥b1+b2? Yes, by adding the two inequalities a1≥b1 and a2≥b2 together. So w1+w2 is in our subset.For 2, we need to show that if α is any scalar, then if w1=(a1,b1,c1) is in our subset, so is αw1. But αw1=(αa1,αb1,αc1). We know since w1 is in our subset that a1≥b1. We need to check that for any α, αa1≥αb1. This inequality is definitely true if α≥0, but we need it to be true for all α, and it's not, because if α is negative, then multiplying the inequality a1≥b1 on both sides by a negative number makes the inequality flip. So we would get αa1≤αb1 if α<0, which means the inequality doesn't hold. Thus, if α<0, then αw1 is not in our subset because it doesn't satisfy αa1≥αb1.So our subset is not a subspace because it doesn't satisfy 2 (it is not closed under scalar multiplication, because any negative scalar would cause this problem).