Okay, We've got a more in depth problem here. Um, so that these vectors negative Four comma, one comma, three, five comma, one comma, six and six comma. Zero comma to show that these do not span riel three dimensional space. But in fact, that they do spin a subspace of three dimensional space consisting of vectors lying only in the plane with the equation X plus 13. Why minus three Z equals zero. All right, so I'm going to write out the three vectors. Vector one is negative for one three. Make sure I don't make any mistakes either. Here, vector to is 516 Vector three is six zero to all. Right, So any vector that can be described would be a constant times Vector one plus a constant times vector to plus a constant times Vector three. So any vector which would be of the form X Why Z would be a constant times. Vector one just negative four one three plus a constant times vector to which is five 16 plus a constant, a different constant times. Vector three, which is six zero to Okay, So now I can write this in matrix form as, um, Wait a minute. I don't want it right in matrix form yet. That's school back a little bit. I want to write. This is negative for times. See, one plus five times C two plus six times C three equals X. And I got that from here. Time. See? One here. Time. See to hear time C three equals that the first coordinate. I can also write that one time. See? One plus one time see too. Plus zero C threes is why. And I can write the re C ones plus 60 twos. Plus to see threes equals Z. I could now rewrite this as an augmented matrix. Negative four five six. Making sure I didn't make any mistake Shut. Um, 110 and, um 362 x. Why Z Let's keep going. So I'm just gonna put one one zero. Why in the first row, Um, then, um, I'm gonna take that and multiply by four and add it to the old first row. So that's going to give us zero. So I'm doing this Times four. Okay, um so and adding it to the first row zero nine six um, X plus four Why? So I'm adding it. I've added it to the other row, which is verified that I did that correctly. Zero nine for six X plus four y. Okay, Now I'm going to take that second row, multiply it times negative three and added to the third row zero one times negative three is negative. Three plus six, his three to. And then this is going to give me Z minus three. Why? Let's keep going. Divide that second row by nine. Okay, Now I'm going to take this. I'm gonna take this, uh, second row and multiply it by negative 1/3 and add it to the third row. Um, okay, I'm gonna multiply it by negative 1/3 added to the third row. Negative 1/3 times. What am I thinking here? Negative 1/3 time. Six is negative. Two negative. 1/3 times. Z minus two. Why is negative? 1/3 Z minus. Think it's into by accident, but it's three. All right, so negative. 1/3 times e minus three. Why equals zero? All right. I'm noting a mistake in what I just did when I was, um, putting these Ah, line two and adding line to tow line three row to to row three. Um, accidentally wrote the wrong value here, So let's go back and try this again. Okay? I was doing negative 1/3 times the second row. That should have been negative 1/3 times the second row. And then I was adding that to the third row. I realized there was a mistake because the equation that I wrote before didn't have accident, and I thought that's weird, especially since the answer that were supposed to get has X in it. So let's just go back and make sure didn't misstate make a mistake. Negative. 3rd 1/3 Time's the second row plus the third row. So now negative 1/3 times X plus, or why plus Z minus three. Why equals zero? So now let's go ahead and simplify that. That's going to give us by multiplying by three on both sides of the equation, or even by negative three. That's going to give us X plus for why minus three Z plus nine. Why equals zero. And now we can combine like terms X plus 13. Why minus three Z equals zero. And that's exactly what we were intended to show that, uh, these three vectors do span this subspace of rial three dimensional space. Um, and that subspace is all the vectors lying in the plane of this equation. They spend that plane.