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. Now I noticed that there are Ford four vectors and it's only three dimensional space. So therefore they are dependent. The set is linearly dependent. Now, I don't think it's obvious to me how these go together, how they're depended on each other. So I'm going to create a matrix notice that I'm writing, um, each vector in column forms. I'm writing the transpose of each vector. Whoops. That should have been to, um 312 negative one negative 11 And I'm going to write an augmented matrix because when you add a constant times all of these vectors together, um, you're going to get zero. That's the definition of linearly dependent. I'm gonna verify that. I wrote everything down correctly because once a few problems ago made a writing mistake, and that takes some time to figure out what was going wrong. All right, well, I'm gonna take negative one times the first row. That's gonna give me 10 Negative. 31 that, uh ah. Zero Okay, Now I'm gonna add the first road in the second row zero to four 00 No, myself. Some more room here. No, I didn't really need more room. Um, now I'm gonna take the first road times, too, and add it to the third row. Zero negative one three times to its six plus two is eight negative. One times two is negative. Two plus one is negative. One. All right, I'm gonna take the second row and divided by two. Now, I'm gonna add that to the third row. Um, two plus eight is 10 zero. All right, now I'm going. Teoh, divide the third row by 10. Now I'm going to take negative too times this and add it to that. It's gonna give me 01 zero negative too. Times this. So that's going to be negative. Two times negative. 1/10 Which is positive. Too tense, which is positive 1/5. Now, I'm going to take, um, three times the third row, the new third row, and I'm gonna add ah, Teoh whips to that. That's going to give me 10 zero, because one times three plus negative. 30 Okay, negative. 3/10 plus one. So that would be 10 tens minus three times. That's 7/10 zero. Now I can write some equations. C one plus 7/10 C four is zero see to plus 1/5 C four is zero C three minus 1/10 c four zero and I'm gonna solve these for C one, C two and C three. C one is 7/10 C four si two is It's negative. 7/10 negative 1/5 c four c three is 1/10 C four. Now I'm going to write that. See one times the first vector. And you know what? I'm just going to write Vector one plus C two times Vector two plus C three times Vector three plus C four times Vector four is zero because they are linearly dependent. But see one is negative. 7/10 C four si two is negative. 1/5 c four c three is 1/10 c four. Not doing anything with this one Since see Ford is not equal zero I can write negative 7/10 times V one minus 1/5 times V two plus 1/10 times V three plus V four equals zero and this should be the zero vector. Um, so let's right the vectors in here now. Negative 7/10 times The first vector, which was negative. One 12 minus 1/5 time's the second vector zero to negative one plus 1/10 times. The third vector plus the last vector. All right, and that's the answer for the dependency, However, let's check in for if it's correct. Um, I'm going to write just for the ah the X coordinate, which has to be zero negative. Seven tense times Negative one is 7/10 plus 3/10 minus one, which is indeed zero. Let's let's try all that. Let's check all the coordinates. Negative 7/10 minus two tents, which I mean two fists, which is 4/10 plus 1/10 minus one zero. That doesn't look right. Negative. 7/10 minus 4/10 plus 1/10 minus one equals 00 that is negative. 11 10th plus one is is negative one, but negative one minus one is is negative. Two. Makes me wonder if I wrote that last vector incorrectly, and that's going to be very annoying. I did not write it incorrectly. So now when I added, Oh, no. When I added row one and row to negative one plus neg. Plus one is zero. Zero plus two is two, um, three plus one is four negative. One plus negative. One is negative. Two. Oh, my goodness. Well, when you make a mistake, you just go back and you fix it. Okay? Now, let's see how this problem affected everything else. All I did here is I divided by to So that's going to give me negative one right there. Okay, Now, I added two rows together. Such that one plus negative one is 08 plus two is 10 and negative too. Plus negative one is negative. Three. Okay, Now, I divided that last row by 10 which would give me negative 3/10. Then that's going to effect this and this. So what I did is I mall applied by three, and I added it to the first row. So negative. Three tense times three is negative. 9/10. And, um, that's not right, either. Negative. Three tense times three is negative. 9/10. But then I add that to the first row, which is just 1 10 10 10 tens minus 9/10 is 1/10. All right, now we're on the right track now. I also multiplied that last row by two and added it to the second row. So negative 3/10. No, I'm all the planet by negative till let's just think of this. Think this through last road times negative Dio would be negative. Two plus two is zero k last row times negative to would be 6/10 which is the same street fifths and 3/5 minus one B 3/5 minus 5/5. It's gonna give us negative too Fifth. Okay, that, of course, changes everything over here. Okay, so let's fix this. 1/10 minus 2/5. Um, minus three tense. So let's fix over here. Negative one times. Positive. 2/5. Positive. 3/10. Okay, now we've got to fix it down here. I have it in three different places, so I'm just going to go ahead and erase and the rays and the rays. Now we can put the new numbers in. The 1st 1 is negative. 1/10. The 2nd 1 is 2/5. The third wine is the re tempts. Let's do our check again. So at least one thing we're learning is that it's it's very important to check your answers. Um X is 1/10 plus 9/10 minus one, which is indeed zero. Why is negative 1/10 Nuss to fist times two. That's four fists is 8/10 plus 3/10 minus one 89 10 11. It's 11 10th minus one would be 10 night, 10 tents minus one is zero good Z negative. 2/10 minus two fists. That's 4/10 plus 6/10 plus one zero. All right. Did I somehow make another mistake? Negative. 2/10 minus 4/10 plus 6/10. Oh, my goodness. At the very beginning, I wrote the wrong term. That's supposed to be one. Oh my! Wow. But wait a minute. I did read it correctly here. So since I wrote it correctly here, that would mean in theory, that's correct. Theory. That's correct. And theory. That's correct. In theory, that would be correct. That's correct. That's correct. That's correct. Okay, I see what I did right here. 2 50 times. Negative one. No, that's correct. 2/5 times negative. One. His negative 2/5 which is negative for tense. The re tense Times two is 6/10 No. All right. And I have found my mistake again. I believe that it's right here. But let's look, um, what I was doing is I was adding this row in this row together, two plus eight is 10 negative. One plus negative one is negative. Two. So now this is going to be negative. Too tense. And now, which is, by the way, negative 1/5. That's now going to effect these. What I'd like to recommend is try not to get too frustrated. This is pretty frustrating to me right now, but try to maintain a positive attitude because everybody makes mistakes. My advice to you. Okay, So you if I multiply that last road times negative too, and add it to the row above it. Negative. Two times one plus two a zero negative, too. Times negative. 1/5 is two fists, two fists plus negative one. His negative. Three fists, 2/5. Try to do this slowly. Okay, I believe that's good. Now I multiplied the last row times three. That's gonna give me negative 3/5. And I have to add that to one to this. All right. That, of course, changes everything. But I am choosing to be positive, believing that this will be the last time. Okay to this minus 3/5. Minus 1/5. That gives us 1/5 the Revis and negative two fists. Now we erase all these things. First one's negative. All the others are positive. Negative. 2/5. All the other is a positive. 3/5 and 1/5 and we try again. But this time I think I learned my lesson. Maybe not all of my lesson, but least some of it. I am going, Teoh. Oops. It didn't work. What just happened? Okay, that was annoying.