00:01
In this video, we're going to show that w is a subspace of rn, where w is the set of all vectors in rn, where ax equals the zero vector, and a is a fixed m by n matrix.
00:17
Now, to show w is a subspace, we want to first show that w is non -empty.
00:23
Now, how do i show w is non -empty? i want to show that the zero vector is in there.
00:28
Well, you know, if i look at the zero vector, and the zero vector is comprised of n components, right? the zero vector is in rn, and a times the zero vector is the zero vector.
00:43
And so that shows zero, the zero vector is in w.
00:48
And so therefore, w is non -empty.
00:51
Now i want to show w is closed under addition and closed under scalar multiplication.
00:58
Now, i'm going to let the vectors x and y belong to w.
01:05
Now, what does it mean to belong to w? that means x and y belong to rn, and if i take a times x, i get the zero vector.
01:17
Additionally, if i take a times y, i also get the zero vector.
01:23
And i want to show that x plus y belongs to w.
01:26
Now, x plus y belongs to rn because rn is a vector space, and vector spaces are closed under vector addition, okay? now, if i look at a times x plus y, i get to distribute the a.
01:44
So, this is a times x plus a times y.
01:49
But, remember x belongs to w, so a times x equals the zero vector.
01:55
And y belongs to w, so a times y is also the zero vector.
01:59
So, ax plus ay is equal to zero plus zero, but zero plus zero is zero.
02:07
So, you get the zero vector back.
02:11
So, x plus y belongs to w, and therefore w is closed under vector addition.
02:20
Now, we want to show that kx belongs to w, where k is a scalar...