Something like this? Why do we even do this? To think about this, let'sĪctually think about it in terms of literally a matrix equation. But now that we have set this up, how do we actually solve You would have the matrix here would be negative 2, 4, 2, negative 5, and this would be negative 6, 7. ![]() Try to represent this right over here as a matrix equation. S and ts in the same order, and you could do that. I would swap the rowsįor the coefficients, but I would still keep the The first row here would be negative 2, 4. Matrix equation with this system, I would just swap all of the rows. The second one first, and I've written the first one second, so this is obviously the same system. so copy and paste, is the same thing as this system, where I'm really just swapping. Just copy and paste it, is the same thing as. For example, you could have, instead of writing it this way, this system is obviously the same thing, is obviously the same thing as. That this contains the same information as that. You get negative 2 times s, negative 2 times s plus 4 times t, 4 times t, is equal to negative 6. Second row and this column, you construct the second equation. Times the second entry, and we add them together, that that must be equal to 7, but when you do that, you essentially construct Times the first entry, the second entry here It's essentially what I just did here, the first entry here The dot product of those, and if you don't know what aĭot product is, don't worry. Row, first column and said, when I take essentially All I did is I multiplied, I dealt with the first This tells us that 2 times s, 2 times s plus negative 5 times t, so I could say minus 5 times t must be equal to the first entry up here, first row, first column, is equal to 7. With this row and that column, so if you think about it, This first row, first column, that's going to be this row. Think about which entries they need to be equal to You say, "Wait, I don't quite get that." If you're saying you don't quite get that, multiply this out. The same constraints on the variable s and t. Vector 7, negative 6, 7, negative 6, is the exact same thing as ![]() That times the column vector, column vector st, s and t, being equal to the column The coefficients here, and I'm going to claim that I'm going to take the coefficients here, so 2, negative 5, 2, negative 5, negative 2, negative 2 and 4, and positive 4. Now what I'm going toĭo is I'm going to take the coefficients here. See or need to appreciate is that this can be representedīy a matrix equation. Bear with me, you willĮnjoy it eventually, what we're about to do, and one day, you will see that Writing computer programs or things like computer programs. Them in appropriate ways if we're, for the most part, Operations, matrix equations to essentially manipulate Matrices are they are ways to represent problems, mathematical problems, ways to represent data, and then we can use matrix That you might see while writing a computer game or while working on some Maybe the left-hand sides are the same, the right-hands keep changing, and this might be something Going to do in this video is that it's very useful in computation where you might solveĪlmost the same system over and over and over again. Through the trouble of it?" The value of what we're It's going to take us more time to this, and you're probably going to say, "Well, why are we even going Represent it esssentially as a matrix equation, and we're going to solve ![]() What we're going to do in this video is represent the same system, but we're going to 2 times 1 plus 5 is 7, and so we have s is equal to 1. That we have over there, is equal to 7, or, and weĬould do this part in our head, 2s must be equal to 2Īnd that s is equal to 1. If t is equal to negative 1, this top equation, youĬould use either one, would simplify to 2 times s. Negative 6 is equal to 1, or you get the t is equal to negative 1. Actually, let's just do it to show how that's relatively straightforward, for at least this example right over here. The left sides of the equations and the right sides of the equations, the s's would cancel out. ![]() Techniques we've used, substitution, elimination, and we could do that right over here. We've seen how to solve this, and there's multiple Voiceover:I have a system of 2 equations with 2 unknowns here.
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