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

Question

(1 point) Let H be the set of all points in the third quadrant in the plane V = R2. That is, H = {(x,y) 1 x H a subspace of the vector space V? 0, y 0). Is 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, 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 on your answers to parts 1-3. choose

Explanation / Answer

a) Well, in general if you want to prove that a subspace H is not empty, then you just have to prove that it contains an element. This element can be the 0 element or any other , so we tend to prove that zero vector belongs to the set and (0,0) lies in this space.

Now, suppose that V is a F vector space, WV, v+wW for every v,wW and uW for every uW and every F. Finally, suppose that you proved that xW for some xV. We must have xW for every F, in particular for =0 we get 0=0xW.

This shows that no matter what you can prove to be inside W, if W is closed under scalar multiplication and addition, then it has to contain 0. Nevertheless, note that very very often, showing that 0W is the simplest way to prove that W.

b) WV, v+wW for every v,wW

This proves that the set which is a subspace is closed under addition as for every v,w belonging to the set W, v+w also belongs to the set as

Ex: (-5,-2) (-3,0)

(-5+(-3),-2+0)=(-8,-2) also belongs to W

c)No, it is not closed under scalar multiplication as uW for every uW and every F

But for an  <0, such as -5(-3,-2)= (15,10) which doesn't belong to the set of W as it is only the third quadrant and 15,10 belongs to the first, hence, it is false and doesn't satisfy this criterion.

d) H is not a subspace of V as it is closed under vector addition but it doesnt satisfy scalar multiplication as the elements of H when multiplied from   such that   belongs to V lies in other quadrants too. Hence, it is not a vector subspace of V,

Following properties needs to be satisfied to qualify as subspace:

Let U V be a subset of a vector space V over F.

Then U is a subspace of V if and only if

1. additive identity: 0 U;

2. closure under addition: u, v U implies u + v U;

3. closure under scalar multiplication: a F, u U implies that au U.

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