00:01
Now we have a set of vectors, and we want to see if they are linearly independent, if the set is linearly independent, or if the set is linearly dependent.
00:15
Now, i notice that there are four vectors, and it's only three -dimensional space, so therefore they are dependent.
00:27
The set is linearly dependent.
00:32
Now, i don't think it's obvious to me how these go together, how they're dependent on each other.
00:45
So i'm going to create a matrix.
00:54
Notice that i'm writing each vector in column form.
01:03
So i'm writing the transpose of each vector.
01:07
Whoops, that should have been two.
01:19
3 -1 -2, negative 1, negative 1, 1.
01:27
And i'm going to write an augmented matrix because when you add a constant times all of these vectors together, you're going to get 0.
01:38
That's the definition of linearly dependent.
01:42
I'm going to verify that i wrote everything down correctly because once a few problems ago, made a writing mistake, and that took some time to figure out what was going wrong.
02:05
All right.
02:09
Well, i'm going to take negative one times the first row.
02:12
That's going to give me 1 -0, negative 3, 1.
02:20
0.
02:20
Okay, now i'm going to add the first row to the second row, 0, 2, 4, 0 .0.
02:34
Now, give myself some more room here.
02:43
Oh, i didn't really need more room.
02:47
Now, i'm gonna take the first row times 2 and add it to the third row.
02:54
0, negative 1.
02:58
3 times 2 is 6 plus 2 is 8.
03:03
Negative 1 times 2 is negative 2 plus 1 is negative 1.
03:14
Alright, i'm going to take the second row and divide it by 2.
03:48
Now i'm going to add that to the third row.
03:59
2 plus 8 is 10.
04:29
Now i'm going to divide the third row by 10.
04:46
Now i'm going to take negative 2.
04:50
Times this and add it to that.
04:54
It's going to give me 0, 1, 0, negative 2 times this.
05:06
So that's going to be negative 2 times negative 1 tenth, which is positive 2 tenths, which is positive 1 5th.
05:19
Now i'm going to take 3 times the third row, the new third row.
05:34
And i'm going to add it to that.
05:44
That's going to give me one, zero, zero, because one times three plus negative three is zero.
05:55
Okay, negative three -tenths plus one.
06:00
So that would be ten -tenths minus three -tenths, that's seven -tenths, zero.
06:06
Now i can write some equations.
06:09
C -1 plus c -10th.
06:12
7 tenths c4 is 0.
06:19
C2 plus 1 5th c4 is 0.
06:27
C3 minus 1 tenth c4 is 0.
06:37
And i'm going to solve these for c1, c2, and c3.
06:40
C1 is 7 tenth c4.
06:48
C2 is.
06:49
It's negative 7 tenths, negative 1 5th, c4.
06:58
C3 is 1 tenth c4.
07:07
So now i'm going to write that c1 times the first vector.
07:16
And you know what? i'm just going to write vector 1 plus c2 times vector 2 plus c3 times vector 2 plus c3 times vector.
07:28
Vector 3 plus c4 times vector 4 is 0 because they are linearly dependent.
07:40
But c1 is negative 7 tenths c4.
07:49
C2 is negative 1 5th c4.
07:59
C3 is 1 tenths c4.
08:05
I'm not doing anything with this one.
08:18
Since c4 does not equal zero, i can write negative 7 tenths times v1 minus 1 fifth times v2 plus one -tenth times v3 plus v4 equals zero.
08:53
And this should be the zero vector.
08:57
So let's write the vectors in here now.
09:00
Negative 7 tenths times the first vector, which is negative 1, 1, 2, minus 1 5th times the second vector, 0 ,2, negative 1, plus 1 tenth times the third vector, plus the last vector.
09:45
All right, and that's the answer for the dependency.
09:49
However, let's check if it's correct.
09:53
I'm going to write just for the x coordinate, which has to be zero.
10:03
Negative 7 tenths times negative 1 is 7 tenths, plus 3 tenths minus 1, which is indeed 0.
10:16
Let's check all the coordinates.
10:19
Negative 7 tenths minus 2 tenths, i mean 2 5ths.
10:28
Which is four tenths, plus one tenths, minus one equals zero.
10:42
That doesn't look right.
10:44
Negative seven tenths minus four tenths plus one tenth minus one equals zero.
11:11
Uh -oh, that is negative 11 tenths plus one is negative one, but negative one minus one is is negative 2.
11:23
Makes me wonder if i wrote that last vector incorrectly, and that's going to be very annoying.
11:34
I did not write it incorrectly.
11:39
So now when i added, oh no, when i added row 1 and 2.
11:50
Negative 1 plus 1 is 0 .0 plus 2 is 2.
12:02
3 plus 1 is 4.
12:05
Negative 1 plus negative 1 is negative 2.
12:09
Oh my goodness.
12:13
Well, when you make a mistake, you just go back and you fix it.
12:21
Okay.
12:23
Now, let's see how this problem affected everything else.
12:33
All i did here is i divided by two, so that's going to give me negative one right there.
12:51
Okay, now i added two rows together such that one plus negative one is zero, 8 plus 2 is 10 and negative 2 plus negative 1 is negative 3.
13:22
Okay? now, i divided that last row by 10, which would give me negative 3 tenths.
13:44
Then that's going to affect this and this.
13:55
So what i did is i multiplied by three, and i added it to the first row.
14:06
So negative three tenths times three is negative nine tenths.
14:19
And that's not right either.
14:31
Negative three tenths times three is negative nine tenths...