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. |