CHAPTER 02.24: VECTORS: How can vectors be used to write simultaneous linear equations? Example

In this segment, we’ll see that how we can use vectors to write simultaneously questions. For example, let’s suppose somebody says “Hey, write the following three equations: 25 x1 plus 5x2 plus x3 equal to 106.8 64 x1 plus 8x2 plus x3 is equal to 177.2 and 144x1 plus 12x2 plus x3 is equal to 279.2 as linear combination of vectors.” So the problem statement is saying - hey given these three simultaneously equations, x1 x2 and x3, are the unknowns, and you would want to write them as linear combination vectors.

So what I want to do first is that is that I want to write down this in the vector form. The left side and the right side. So we are able to see the left side is equal to the right side of each of these three equations; we want to make each of the left sides, each of the three left sides, as components of this vector right here. The next component is going to be 64 x1 plus 8 x2 plus x3 and the next component is going to be 144 x1 plus 12 x2 plus x3. So I got these three components in my left matrix right here, and if I want this particular vector - this vector - I want this vector to be the same as what’s here, then I’ll simply put the writing sides here 106.8 177.2 and 279.2. So that makes this vector to be same as this vector as this vector as based on these three questions right here. But I can further simplify by breaking this down as the linear combination of three vectors I can say x1 is right here so I got 25 64 and 144 right here then x2 is right here and I got 5 8 and 12 right here and I got x3 right here which is 1 1 and 1 right here and that will be equal to 106.8 177.2 and 279.2.

So now what I’ve done is I have taken three vectors, with these three numbers and then these three numbers and 1 1 1 right here. So I’ve taken three vectors right here: 1 2 and 3 multiply by some scale of x1 x2 x3 and equal to another vector which is the one right here. So that’s how I have written these equations of linear combination of three vectors is equal to another vector. That’s the end of the segment.