CHAPTER 02.23: VECTORS: How can vectors be used to write simultaneous linear equations?

 

 

In this segment, weíre going to see how we can use vectors to write simultaneous linear equations. So letís suppose somebody gives us m linear equations where the equations with n unknowns. So somebody gives us m linear equations with n unknowns. So letís suppose those are like this: a11(x1)+a12(x2)+Ö +a1n(xn)=c1. And then we go to the last equation which will be - because we are given m in your equation - will be am1(x1)+am2(x2)+Ö+amn(xn)=cm. So we are given these m equations and we have n unknowns. And how can we write these in their vector form you very well see is what I can do is as follows I can say that they I want to write this one as the Ė Iím going to put one vector equal to another vector thatís what Iím gonna do first and I want in this vector what im gonna do is Iím going to put these as the entries Iím gonna put [a11(x1)+a12(x2)+Ö +a1n(xn)] as the first entry and then now my second entry third entry and the the mth entry will be [am1(x1)+am2(x2)+Ö+amn(xn)] and then here ill have c1 all the way up to cm. So thatís what Iíll have there.

 

So what I have basically done is that somebody gave me m simultaneous linear equations and Iím now putting them into vector form where this is one column vector is equal to another column vector, which I simply get from because those are the right sides of those equations. But you can very well see that I can break it down further by saying as follows: that I will put down in my first column as a11 all the way up to am1 and Iím going to multiply it by x1 because all of these are multiplied by x1. So Iím going to put my a11 a21 and so on and so forth up to am1 and multiply by x1 then I can add x2 and those will also be multiplied by a vector which has m entries in it. So it will be a12 all the way up to am2 and I can keep on doing this till I have xn. Thatíll get multiplied by a1n all the way up to m entries. There amn thatíll be equal to c1 all the way up to cm. So what are you finding out is that if I call this to be - letís suppose the A1 vector - Iíll call this to be the A2 vector. Theyíre all column vectors; I call this to be the A3 vector and I call this to be my C vector.

 

What Iím finding out is that - hey Iím going to X1(A1)+X2(A2)Ö. +Xn(An), I shouldnít call this A3. I should call this A. It would be An. An is equal to the C vector so what you are finding out here is that Iím getting a linear combination of n vectors multiplied by X1, X2, all the way to Xn equal to C vector. So what that means - that if I take a linear combination of these n vectors and I put it equal to C vector and find if Iím able to find the solution to this linear combination of these n vectors equal to C vector, then I have found the solution because thatís what Iím looking for. Iím looking for the values of X1, X2 all the way to Xn. So when youíre looking up this set of equations right here what I forgot to mention was that everything is known here including the coefficients a11, a12 and so on and so forth and these right hand sides. The unknowns are simply the x values X1, X2 all the way up to Xn. So if we are able to find the combination for the scalers which satisfy this linear combination equal to the C vector, we have found the solutions. So thatís how we write; we can use vectors to write simultaneous linear equations. And that is the end of this segment