00:01
In this question, they're giving that u and w are subspaces of v.
00:06
And we introduce a new set which is called u plus w.
00:10
Basically, it consists of sum of elements from u and w.
00:16
And yeah, u belongs to u, you take you from u, w from w, and then you add them.
00:23
That's how you get the set u plus w.
00:25
And we're asked to show that u plus w is a subspace of v.
00:29
So to do that, we need to check three conditions.
00:34
First, we want to show that the zero vector is in u plus w.
00:40
Second, we want to show that it's closed under addition, meaning for any u, well, let's call them, for any z1, z2 in u plus w, there are some is also in u plus w.
01:07
Finally, we want to show closeness under scolar multiplication, meaning for any number lambda and any vector z in u plus w, we want to show that lambda z is also in w in u plus w.
01:26
Let's check the first condition.
01:30
So since you, both u and w are subspaces, zero vector belongs to you and you...