00:01
In this question, we are given a set of vectors and we are asked to show that this set is linearly dependent by giving a linear combination which gives 0.
00:15
So, recall that a linear combination of the vectors is, so let the first vector here be as 1, second vector b .s2, the third vector bs 3.
00:29
So let's construct, create a linear combination of these three vectors.
00:34
C1s1 plus c2 as 2 plus c3 as 3 and we want this to be equal to 0, right? where c1, c2 and c3 are any real numbers.
00:51
Might be any real numbers.
00:55
Now, a set is linearly independent.
00:59
If this equation here is only true when all see when each c1, c2 and c3 are equal to zero and only for zero, right? no other linear combination gives you, if no other linear combination gives you zero, then vectors are called linearly independent.
01:22
Otherwise, if you can find c1, c2 and c3, which are not zero at the same, time and this linear combination is zero, then these vectors are going to be linearly dependent.
01:39
So therefore, in this question they're basically asked to find the numbers c1, c2, and c3, not zero at the same time, and so that the sum is zero.
01:55
Some c1 s1 plus c2 as 2 plus c3 is equal to 0.
02:00
To do that, we're just going to do it like by guessing.
02:09
So let's write down the linear combination explicitly.
02:30
And let's try to guess numbers c1, c2, and c3 so that in the sum we get zero in each component.
02:42
Well, to get zero in the first component, it seems like if you plug in c1 equals 1, c2 equals 1 and c3 equals negative 1, it seems like they're going to get zero in the first component.
02:55
Let's see if this is going to work for all other components.
03:01
So c1 equals c2 equals 1.
03:04
C3 equals negative 1.
03:09
So we will get 1, 2, 3, 3, 4 plus c2 we are also going to replace by 1.
03:18
We are going to get 1 0 ,0, 1.
03:26
And c3 we are going to replace by negative 1.
03:29
So we are going to get minus 268, 10.
03:39
Now, in the first entry, 1, we will get 1.
03:45
Plus 1 minus 2 which is 0.
03:47
In the second entry we are going to get 2 plus 0 minus 6 and that's not 0.
03:55
So this linear combination is not working.
03:58
So because we are not getting 0.
04:04
You are getting negative 4 in the third component and we are going to get negative 4 in the third component in the last component, 4th component.
04:13
So this didn't work.
04:14
This is not 0.
04:20
Let's try something else.
04:25
Or the rhythm easier way...