00:01
Let v equal r3, the space, and let h be the subset of v of all points on the plane 5x plus 7y plus 7z equal 35.
00:13
The question is whether h is a subspace of the vector space v.
00:21
That we're going to do in three steps, we get to answer each of those.
00:26
First step, does h contain the zero vector of v? second part, is h closed under addition? third part, is h closed under scalar multiplication? and fourth and last part, is h a subspace of vector space v? good, so let's start with one.
00:57
Does h contain zero vectors? so first we remark here that h, the subset of r3, is defined as the elements in r3, that is x, y, z in r3, such that 5x plus 7y plus 7z equals 35.
01:26
This is the formal definition of h.
01:30
Of course, this is the subset of r3.
01:35
So we want to know if the zero vector, 0, 0, 0, verified this equation.
01:45
It's clear that it's not the case here.
01:48
So we can say that 0, 0, 0 does not belong to h.
01:56
Since 5 times the x component of 0, 0, 0, which is 0, plus 7 times the second component, 0, plus 7 times the third component, which is also 0, give us 0, which is different from 35, which is the condition given here.
02:19
So 0, 0, 0 is not in h.
02:22
Good, so you got to choose the answer is not in h.
02:29
Now let's see part two.
02:31
Is h closed under addition? the answer is no.
02:35
H is not closed under addition.
02:50
And for that we give two vectors which are in h, but such that their sum is not in h.
03:00
In this case, it's easy.
03:01
You can find 0, 0, 5, and 0, 5, 0 are in h.
03:14
That is because if you do this calculation here, defining the elements in h, you see that 5 times 0, plus 7 times 0, plus 7 times 5 is 35.
03:30
And the same thing here because we have 7 times y.
03:34
The other two are zeros.
03:35
So it's clear that these two elements are in h, but the sum of these two vectors, which we know is, if we add together these two vectors, we get the vector 0, 0 plus 5, 5 is in the second component, and the third component, 5 plus 0 is 5.
04:02
But this vector is not in h, since 5 times the first component of this vector, 0, plus 7 times the second component, 5, plus 7 times the third component, 5, give us 35 plus 35, which is 7t, that is, is not equal to 35, which is the condition required for a vector to be in h.
04:41
So we have found two vectors in h, such that their sum is not in h, and so h is not closed under addition.
04:50
So you get to enter these two vectors.
04:57
You enter like this, 0, 0, 5, comma, to separate the vectors, then, 0, 5, 0.
05:12
That's what you get to enter in the text box...