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In the last video we saw how a matrix and figuring out its

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inverse can be used to solve a system of equations.

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And we did a 2 by 2.

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And in the future, we'll do 3 by 3's.

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We won't do 4 by 4's because those take too long.

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But you'll see it applies to kind of an n by n matrix.

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And that is probably the application of matrices that

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you learn if you learn this in your Algebra 2 class, or your

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Algebra 1 class.

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And you often wonder, well why even do the

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whole matrix thing?

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Now I will show you another application of matrices that

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actually you're more likely to see in your linear algebra

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class when you take it in college.

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But the really neat thing here is, and I think this will

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really hit the point home, that the matrix representation

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is just one way of representing multiple types of

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problems. And what's really cool is that if different

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problems can be represented the same way, it kind of tells

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you that they're the same problem.

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And that's called an isomorphism in math.

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That if you can reduce one problem into another problem,

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then all the work you did with one of them

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applies to the other.

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But anyway, let's figure out a new way that

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matrices can be used.

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So I'm going to draw some vectors.

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Let's say I have the vector-- Let's call this vector a.

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And I'm going to just write this is as a column vector.

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And all of this is just convention.

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These are just human invented things.

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I could have written this diagonally.

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I could have written this however.

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But if I say vector a is 3, negative 6.

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And I view this as the x component of the vector, and

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this is equal to the y component of the vector.

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And then I have vector b.

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Vector b is equal to 2, 6.

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And I want to know are there some combinations of vectors a

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and b-- where you can say, 5 times vector a, plus 3 times

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vector b, or 10 times victor a minus 6 times vector b-- some

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combination of vector a and b, where I can get vector c.

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And vector c is the vector 7, 6.

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So let me see if I can visually draw this problem.

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So let me draw the coordinate axes.

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Let's see this one.

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3, negative 6.

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That'll be in quadrant-- these are both

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in the first quadrant.

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So I just want to figure out how much of the

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axes I need to draw.

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So let's see-- Let me do a different color.

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That's my y-axis.

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I'm not drawing the second or third quadrants, because I

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don't think our vectors show up there.

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And then this is the x-axis.

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Let me draw each of these vectors.

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So first I'll do vector a.

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That's 3, negative 6.

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1, 2, 3, and then negative 6.

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1, 2, 3, 4, 5, 6.

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So it's there.

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So if I wanted to draw it as a vector, usually

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start at the origin.

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And it doesn't have to start at the origin like that.

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I'm just choosing to.

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You can move around a vector.

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It just has to have the same orientation

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and the same magnitude.

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So that is vector a for the green.

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Now let me do in magenta, I'll do vector b.

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That is 2, 6.

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1, 2, 3, 4, 5, 6.

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So 2, 6 is right over there.

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And that's vector b.

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So it'll look like this.

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That's vector b.

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And then let me write down vector a down there.

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That's vector a.

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And I want to take some combination of

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vectors a and b.

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And add them up and get vector c.

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So what does vector c look like?

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It's 7, 6.

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Let me do that in purple.

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So 1, 2, 3, 4, 5, 6, 7.

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Comma 6.

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So 7, 6 is right over there.

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That's vector c.

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Vector c looks like that.

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I'm going to draw it like that.

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And that's vector c.

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So what was the original problem I said?

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I said I want to add some multiple of vector a to some

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multiple of vector b, and get vector c.

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And I want to see what those multiples are.

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So let's say the multiple that I multiply

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times vector a is x.

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And the multiple of vector b is y.

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So I essentially want to say that-- let me do it in another

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neutral color-- that vector ax-- that's how much of vector

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a I'm contributing-- plus vector by-- that's how much of

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vector b I'm contributing-- is equal to vector c.

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And you know, maybe I can't.

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Maybe there's no combinations of vector a and b when you add

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them together equal vector c.

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But let's see if we can solve this.

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So how do we solve?

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So let's expand out vectors a and b.

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Vector a is what?

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3, negative 6.

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So vector a, we could write as 3, minus 6 times x.

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That just tells us how much vector a we're contributing.

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Plus vector b, which is 2, 6.

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And then y is how much vector b we're contributing.

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And that is equal to 7, 6.

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Vector c.

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Now this right here, this problem can be rewritten just

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based on how we've defined matrix multiplication, et

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cetera, et cetera, as this.

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As 3, minus 6, 2, 6, times x, y, is equal to 7, 6.

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Now how does that work out?

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Well think about how matrix multiplication works out.

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The way we learned matrix multiplication, we said, 3

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times x, plus 2 times y is equal to 7.

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3 times x plus 2 times y is equal to 7.

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That's how we learned matrix multiplication.

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That's the same thing here.

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3 times x, plus 2 times y, is going to be equal to 7.

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These x and y here are just scalar numbers.

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So 3 times x plus 2 times y is equal to 7.

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And then matrix multiplication here, minus 6 times x plus 6

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times y is equal to 6.

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That's just traditional matrix multiplication that we learned

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several videos ago.

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That's the same thing here.

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Minus 6x plus 6y is equal to 6.

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These x's and y's are just numbers.

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They're just scalar numbers.

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They're not vectors or anything.

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We would just multiply them times both of these numbers.

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So hopefully you see that this problem is the exact same

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thing as this problem.

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And you've maybe had an a-ha moment now, if you watched the

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previous video.

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Because this matrix also represented the problem, where

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do we find the intersection of two lines?

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Where the two lines-- I'm just going to do it on the side

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here-- the intersection of the two lines, 3x plus

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2y is equal to 7.

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And minus 6x plus 6y is equal to 6.

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And so, I had drawn two lines.

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And we said, what's the point of intersection,

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et cetera, et cetera.

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And it was represented by this problem.

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But here, we have-- well I won't say a completely

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different problem, because we're learning they're

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actually very similar-- but here I'm doing a problem of,

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I'm trying to find what combination of the matrices a

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and b add up to the matrix c.

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But it got reduced to the same matrix representation.

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And so we can solve this the same exactly way we solved

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this problem.

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If we call this the matrix a, let's figure out a inverse.

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So we get a inverse is equal to what?

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It equals 1 over the determinant of a.

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The determinant of a is 3 times 6.

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18 minus minus 12.

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So that's 18 plus 12, which is 1/30.

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And we did this in the previous video.

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You swap these two numbers.

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So you get 6 and 3.

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And then you make these two negatives.

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So you get 6 and minus 2.

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That's a inverse.

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And now to solve for x and y, we can multiply both sides of

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this equation by a inverse.

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If you multiply a inverse times a, this cancels out.

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So you get x, y is equal to a inverse times this.

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It's equal to 1/30 times 6, minus 2, 6, 3.

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Times 7, 6.

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And remember, with matrices, the order that

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you multiply matters.

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So on this side, we multiplied a inverse on

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this side of the equation.

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So we have to do a inverse on the left side on this side of

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this equation.

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So that's why did it here.

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If we did it the other way, all bets are off.

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So what does this equal?

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This is equal to 1/30 times-- and we did this the previous

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problem-- 6 times 7 is 42, minus 12.

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30.

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6 times 7, 42.

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Plus 18.

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60.

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So that equals 1, 2.

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So what does this tell us?

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This tells us that if we have 1 times vector a, plus 2

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times vector b.

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1 times-- this is 1-- and 2 times vector b.

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So 1 times vector a plus 2 times vector b is

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equal to vector c.

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And let's confirm that visually.

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So 1 times vector a.

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Well that's vector a right there.

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So if we add 2 vector b's to it, we should get vector c.

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So let's see if we can do that.

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So if we just shift vector b over this way, well vector

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let's see, vector b is over 2 and up 6.

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So over 2 and up 6 would get us there.

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So 1, vector b-- just doing heads to tail visual method of

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adding vectors-- would get us there.

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1, 2, 3.

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Good.

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No, let me see.

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1, 2, 3.

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And then vector b goes over two more.

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two more.

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So it'll get us up 6.

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It's like that.

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So that's 1, vector b.

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And then if we add another-- but we want 2 times vector b.

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We essentially need two vector b's.

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So we had one, and then we add another one.

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I think visually you see that it does actually-- I didn't

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want to do it like that.

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I wanted to use the line tool so it looks neat.

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So you add another vector b.

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And there you have it.

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That's a vector b.

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So it's 2 times vector b.

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So it's the same direction as vector b, but it's two times

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the length.

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So we visually showed it.

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We solved it algabraically.

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But the real learning, and the big real discovery of this

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whole video, is to show you that the matrix representation

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can represent multiple different problems. This was a

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finding the combinations of a vector problem.

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And the previous one it was figure out if

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two lines can intersect.

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But what it tells you is that these two problems are

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connected in some deep way.

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That if we take the veneer of reality, that underlying it,

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they are the same thing.

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And frankly, that's why math is so interesting.

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Because when you realize that two problems are really the

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same thing, it takes all of the superficial human veneer

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away from things.

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Because our brains are kind of wired to perceive the world in

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a certain way.

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But it tells us that there's some fundamental truth,

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independent of our perception, that is tying all of these

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different concepts together.

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But anyway, I don't want to get all mystical on you.

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But if you do see the mysticism in

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math, all the better.

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But hopefully you found that pretty interesting.

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And actually, I know I'm going over time, but I think this

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is-- A lot of people take linear algebra, they learn how

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to do all of the things, and they say, well what is the

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whole point of this?

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But this is kind of an interesting

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thing to think about.

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We had this had vector a and we had this vector b.

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And we were able to say, well there's some combinations of

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the vectors a and b, that when we added it up,

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we got vector c.

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So an interesting question is, what are all the vectors that

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I could get to by adding combinations of

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vectors a and b.

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Or adding or subtracting.

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Or you could say, I could multiply

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them by negative numbers.

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But either way.

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What are all of the vectors that I can get by taking

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linear combinations of vectors a and b?

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And that's actually called the vector space spanned by the

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vectors a and b.

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And we'll do more of that in linear algebra.

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And here we're dealing with a two

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dimensional Euclidean space.

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We could have had three dimensional vectors.

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We could've had n dimensional vectors.

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So it gets really, really, really abstract.

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But this is, I think, a really good toe dipping for linear

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algebra as well.

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So hopefully I haven't confused or overwhelmed you.

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And I'll see you in the next video.

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