 In this video, we're gonna provide the solution for question number 15 on the practice midterm exam, number one for math 2270. Let Q be the first quadrant in the XY plane. That is, Q is the set of order pairs X comma Y where X is not negative and Y is not negative. We wanna show that Q is not a subspace of R2. Now, if it were a subspace, there's three things that have to be satisfied. First, zero would have to belong to the subset, which actually is true, right? The way that we've defined it, X could equal zero and Y could equal zero. So we see that the zero vector does belong to Q. So the first action to actually satisfy. The second axiom is, if we have two vectors in the subset, then their sum must also be in there. Well, if you have a vector whose X coordinate is not negative, and you have two X coordinates whose X coordinate, if you have two vectors whose X coordinate R is not negative, their sum will still be non-negative. And the same is also true for Y. So this set actually is closed under addition. So if it's not a subspace, it's gotta be the third axiom. There's something wrong with scalar multiplication. If I take a vector that's in Q and I scale it by some real number, do I get something outside of Q in? So note here. So note that we can take the vector one, one, which does belong to Q and take the scalar negative one, which is a real number. But if I take negative one and I times that by one, one, this gives me the vector negative one, negative one, which actually does not belong to Q because Q has to have non-negative entries and negative one, negative one in fact has negative entries. And so therefore we can say Q is not a subspace. Now there's three things required to be a subspace to show that it's not a subspace. We only have to provide a counter example to one of the three axioms. We don't have to destroy all three of them because in fact this one passed the other two. We just have to show that this subset is not closed under scalar multiplication.