00:01
So if we want to determine if it is possible to have or w is a linear combination of these three vectors, what that's really saying is, is there some number that we can multiply v1, v2, and v3 by? when we add them up, we get w.
00:18
So this, if we were to go ahead and just kind of multiply things out, you'll see is actually a good use of us trying to do a matrix to solve this, because it would just turn into a system of equations.
00:33
So let's just go ahead and do this really quickly.
00:36
So we have c1 times 3 -1 -8 plus c -2.
00:47
Here, let me scoot these over a little bit.
00:51
So then c -2 times 501 plus c -3, 1 -3 -3 -3 -3 -9 -1.
01:03
Like that.
01:05
So now we can go ahead and distribute the c -1, c -2 and c -3.
01:10
To each of the vectors.
01:14
So it'd be 3c1, negative 1, c1, 8c1, plus 5c2.
01:23
And i know c2 times 0 is just going to be 0, but i'm going to write it as 0c2.
01:29
And then we have just 1c2 here.
01:35
And then plus.
01:36
So we'd have 1c3, 3, c3, and negative 1, c3.
01:46
And then remember when we add vectors together, we can just do it component -wise.
01:52
So we just add straight across, straight across, straight across.
01:56
So the top is going to be 3c1 plus 5, c2, plus c3, all over negative 1, c1, i almost forgot my 1 here.
02:13
I mean, it doesn't really matter, but just kind of for a completion sake.
02:19
0c2 plus 3c3, 8c1 plus 1c2, minus 1c3.
02:31
And this is supposed to be equal to w, and w, actually here, let me write it on this side, is going to be, sorry, w here, 3, 9, 18.
02:43
So if we were to just go ahead and drop the matrix brackets here, and instead of just having 1 equals, you might see now how this turns into an actual question involving three variables.
03:11
So just kind of going back to how we had it before.
03:20
So then we can go ahead and pull out the c1, c2, c3.
03:24
And then i actually write in the standard matrix form.
03:27
So it would be 351, negative 1, 0, 3 ,8, 1, negative 1, negative 1, and then this is going to be times c1, c2, c3.
03:44
And it's still going to be equal to 3, 9, 18.
03:49
So if we solve this matrix here and do our row reduction and all that, this should go ahead and tell us what c1, c2, and c3 are going to be.
04:02
So now i'm just going to write this as the augmented matrix and not write c1, c2, c3, and we just have to keep in mind this is c1, c2, and c3.
04:11
So we have 3, 5, 1, then we put the little line here for that equal sign, then close off our matrix, negative 1, 03, 9, 8, 1, negative 1, 18.
04:27
So now our goal is to try to get ones along our diagonal here.
04:40
And just to make our lives a little bit easier, like i don't want to have to divide by one -third.
04:45
So i'm going to switch rows two and one.
04:49
So i'm going to do row one, switched with row two.
04:59
And you might use a slightly different notation for like saying how, you do your steps, but this is just kind of how i learned how to do it.
05:10
So all we're going to do is just take row two, put it up on row one.
05:14
So negative 1, 0, 3, 9.
05:17
Let me go ahead and close this.
05:20
And then now row 1 becomes row 2.
05:24
So 3, 5, 1, 3 .1, 3.
05:27
And then 8, 1, negative 1, 18.
05:33
So now we can go ahead and clear the rows below this negative 1 here.
05:42
So it looks like we would want to multiply row 1 by 3 and then add that to row 2.
05:55
And we could also multiply row 1 by 8 and add that to row 3.
06:03
And then notice that 3 and the 3 here get cancelled out because negative 3 or negative 1 times 3, negative 3 plus 3, 0, negative 1 times 8.
06:14
Negative 8 plus 8 0 so let's go ahead and do that so the top row stays the same we didn't touch that at all so it's negative 1 0 3 9 and then for the second row so that's 0 3 times 0 plus 5 is 5 3 plus 1 is 10 and then 3 times 9 plus 3 is 30 and then for the 6.
06:52
And then for the 6 second row so that was zero then zero times eight plus one is one three times eight um it's 24 plus negative one would be negative or not negative would be 23 and then nine times eight is 72 and 72 plus 18 is 90 so now at this point there's two we can do.
07:30
Notice how this second row has all multiples of five.
07:35
So we could just divide the entire row by five and then try to cancel out that one there.
07:43
And then notice that's already canceled out above, so we don't have to do anything with row one...