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## Solving equations with inverse matrices

# Matrix word problem: vector combination

## Video transcript

In the last video we saw how a
matrix and figuring out its inverse can be used to solve
a system of equations. And we did a 2 by 2. And in the future, we'll
do 3 by 3's. We won't do 4 by 4's because
those take too long. But you'll see it applies to
kind of an n by n matrix. And that is probably the
application of matrices that you learn if you learn this in
your Algebra 2 class, or your Algebra 1 class. And you often wonder,
well why even do the whole matrix thing? Now I will show you another
application of matrices that actually you're more likely to
see in your linear algebra class when you take
it in college. But the really neat thing here
is, and I think this will really hit the point home, that
the matrix representation is just one way of representing
multiple types of problems. And what's really
cool is that if different problems can be represented the
same way, it kind of tells you that they're the
same problem. And that's called an isomorphism
in math. That if you can reduce one
problem into another problem, then all the work you
did with one of them applies to the other. But anyway, let's figure
out a new way that matrices can be used. So I'm going to draw
some vectors. Let's say I have the vector--
Let's call this vector a. And I'm going to just write this
is as a column vector. And all of this is
just convention. These are just human
invented things. I could have written
this diagonally. I could have written
this however. But if I say vector a
is 3, negative 6. And I view this as the x
component of the vector, and this is equal to the y component
of the vector. And then I have vector b. Vector b is equal to 2, 6. And I want to know are there
some combinations of vectors a and b-- where you can say, 5
times vector a, plus 3 times vector b, or 10 times victor a
minus 6 times vector b-- some combination of vector a and b,
where I can get vector c. And vector c is the
vector 7, 6. So let me see if I can visually
draw this problem. So let me draw the
coordinate axes. Let's see this one. 3, negative 6. That'll be in quadrant--
these are both in the first quadrant. So I just want to figure
out how much of the axes I need to draw. So let's see-- Let me do
a different color. That's my y-axis. I'm not drawing the second or
third quadrants, because I don't think our vectors
show up there. And then this is the x-axis. Let me draw each of
these vectors. So first I'll do vector a. That's 3, negative 6. 1, 2, 3, and then negative 6. 1, 2, 3, 4, 5, 6. So it's there. So if I wanted to draw it
as a vector, usually start at the origin. And it doesn't have to start
at the origin like that. I'm just choosing to. You can move around a vector. It just has to have the
same orientation and the same magnitude. So that is vector
a for the green. Now let me do in magenta,
I'll do vector b. That is 2, 6. 1, 2, 3, 4, 5, 6. So 2, 6 is right over there. And that's vector b. So it'll look like this. That's vector b. And then let me write down
vector a down there. That's vector a. And I want to take some
combination of vectors a and b. And add them up and
get vector c. So what does vector
c look like? It's 7, 6. Let me do that in purple. So 1, 2, 3, 4, 5, 6, 7. Comma 6. So 7, 6 is right over there. That's vector c. Vector c looks like that. I'm going to draw
it like that. And that's vector c. So what was the original
problem I said? I said I want to add some
multiple of vector a to some multiple of vector b,
and get vector c. And I want to see what
those multiples are. So let's say the multiple
that I multiply times vector a is x. And the multiple of
vector b is y. So I essentially want to say
that-- let me do it in another neutral color-- that vector ax--
that's how much of vector a I'm contributing-- plus vector
by-- that's how much of vector b I'm contributing--
is equal to vector c. And you know, maybe I can't. Maybe there's no combinations of
vector a and b when you add them together equal vector c. But let's see if we
can solve this. So how do we solve? So let's expand out
vectors a and b. Vector a is what? 3, negative 6. So vector a, we could write
as 3, minus 6 times x. That just tells us how much
vector a we're contributing. Plus vector b, which is 2, 6. And then y is how much vector
b we're contributing. And that is equal to 7, 6. Vector c. Now this right here, this
problem can be rewritten just based on how we've defined
matrix multiplication, et cetera, et cetera, as this. As 3, minus 6, 2, 6, times
x, y, is equal to 7, 6. Now how does that work out? Well think about how matrix
multiplication works out. The way we learned matrix
multiplication, we said, 3 times x, plus 2 times
y is equal to 7. 3 times x plus 2 times
y is equal to 7. That's how we learned matrix
multiplication. That's the same thing here. 3 times x, plus 2 times y, is
going to be equal to 7. These x and y here are
just scalar numbers. So 3 times x plus 2 times
y is equal to 7. And then matrix multiplication
here, minus 6 times x plus 6 times y is equal to 6. That's just traditional matrix
multiplication that we learned several videos ago. That's the same thing here. Minus 6x plus 6y
is equal to 6. These x's and y's are
just numbers. They're just scalar numbers. They're not vectors
or anything. We would just multiply them
times both of these numbers. So hopefully you see that this
problem is the exact same thing as this problem. And you've maybe had an a-ha
moment now, if you watched the previous video. Because this matrix also
represented the problem, where do we find the intersection
of two lines? Where the two lines-- I'm just
going to do it on the side here-- the intersection of
the two lines, 3x plus 2y is equal to 7. And minus 6x plus 6y
is equal to 6. And so, I had drawn two lines. And we said, what's the
point of intersection, et cetera, et cetera. And it was represented
by this problem. But here, we have-- well I
won't say a completely different problem, because
we're learning they're actually very similar-- but here
I'm doing a problem of, I'm trying to find what
combination of the matrices a and b add up to the matrix c. But it got reduced to the same
matrix representation. And so we can solve this the
same exactly way we solved this problem. If we call this the matrix a,
let's figure out a inverse. So we get a inverse
is equal to what? It equals 1 over the
determinant of a. The determinant of
a is 3 times 6. 18 minus minus 12. So that's 18 plus 12,
which is 1/30. And we did this in the
previous video. You swap these two numbers. So you get 6 and 3. And then you make these
two negatives. So you get 6 and minus 2. That's a inverse. And now to solve for x and y, we
can multiply both sides of this equation by a inverse. If you multiply a inverse times
a, this cancels out. So you get x, y is equal to
a inverse times this. It's equal to 1/30 times
6, minus 2, 6, 3. Times 7, 6. And remember, with matrices,
the order that you multiply matters. So on this side, we multiplied
a inverse on this side of the equation. So we have to do a inverse on
the left side on this side of this equation. So that's why did it here. If we did it the other way,
all bets are off. So what does this equal? This is equal to 1/30 times--
and we did this the previous problem-- 6 times 7
is 42, minus 12. 30. 6 times 7, 42. Plus 18. 60. So that equals 1, 2. So what does this tell us? This tells us that if we have
1 times vector a, plus 2 times vector b. 1 times-- this is 1-- and
2 times vector b. So 1 times vector a plus
2 times vector b is equal to vector c. And let's confirm
that visually. So 1 times vector a. Well that's vector
a right there. So if we add 2 vector b's to
it, we should get vector c. So let's see if we
can do that. So if we just shift vector b
over this way, well vector let's see, vector b is
over 2 and up 6. So over 2 and up 6 would
get us there. So 1, vector b-- just doing
heads to tail visual method of adding vectors-- would
get us there. 1, 2, 3. Good. No, let me see. 1, 2, 3. And then vector b goes
over two more. two more. So it'll get us up 6. It's like that. So that's 1, vector b. And then if we add another-- but
we want 2 times vector b. We essentially need
two vector b's. So we had one, and then
we add another one. I think visually you see that
it does actually-- I didn't want to do it like that. I wanted to use the line
tool so it looks neat. So you add another vector b. And there you have it. That's a vector b. So it's 2 times vector b. So it's the same direction as
vector b, but it's two times the length. So we visually showed it. We solved it algabraically. But the real learning, and the
big real discovery of this whole video, is to show you that
the matrix representation can represent multiple different
problems. This was a finding the combinations
of a vector problem. And the previous one it
was figure out if two lines can intersect. But what it tells you is that
these two problems are connected in some deep way. That if we take the veneer of
reality, that underlying it, they are the same thing. And frankly, that's why math
is so interesting. Because when you realize that
two problems are really the same thing, it takes all of the
superficial human veneer away from things. Because our brains are kind of
wired to perceive the world in a certain way. But it tells us that there's
some fundamental truth, independent of our perception,
that is tying all of these different concepts together. But anyway, I don't want to
get all mystical on you. But if you do see
the mysticism in math, all the better. But hopefully you found that
pretty interesting. And actually, I know I'm going
over time, but I think this is-- A lot of people take linear
algebra, they learn how to do all of the things, and
they say, well what is the whole point of this? But this is kind of
an interesting thing to think about. We had this had vector a and
we had this vector b. And we were able to say, well
there's some combinations of the vectors a and b, that
when we added it up, we got vector c. So an interesting question is,
what are all the vectors that I could get to by adding
combinations of vectors a and b. Or adding or subtracting. Or you could say,
I could multiply them by negative numbers. But either way. What are all of the vectors
that I can get by taking linear combinations of
vectors a and b? And that's actually called the
vector space spanned by the vectors a and b. And we'll do more of that
in linear algebra. And here we're dealing
with a two dimensional Euclidean space. We could have had three
dimensional vectors. We could've had n dimensional
vectors. So it gets really, really,
really abstract. But this is, I think, a really
good toe dipping for linear algebra as well. So hopefully I haven't confused
or overwhelmed you. And I'll see you in
the next video.