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Â Now, so far we haven't really talked about the coordinate system of our vector space,

Â the coordinates in which all of our vectors exist.

Â But it turns out in doing this thing of projecting,

Â of taking the dot product, we are projecting our vector onto one which

Â we might use as part of a new definition of the coordinate system.

Â So in this video we'll look at what we mean by coordinate systems and

Â we'll do a few cases of changing from one coordinate system to another.

Â So, remember that a vector, like this guy r here if just an object that takes us

Â from the origin to some point in space.

Â Which could be some physical space or it could be some data space,

Â like bedrooms and thousands of Euros for the house or something like that.

Â What we haven't talked about so

Â far really is the coordinate system that we use to describe space.

Â So we could use a coordinate system defined by these two vectors here,

Â I'm going to give them names, we called them i and j before,

Â I'm going to give them names e1 and e2.

Â I'm going to define them to be of unit lengths, so

Â I'm going to give them a little hat, meaning they're of unit length, and

Â I'm going to define them to be the vectors 1,0, and 0,1.

Â And if I had more dimension in my space, I could have e3 hat, e4 hat,

Â e5 hat, e 1 million hat, whatever.

Â Here the instruction then is that r is going to be equal

Â to doing a vector sum of 2e1 or 3e1 and then some number of e2.

Â So we'll call it to 3e1 hats plus 4e2 hats.

Â And so we'll write it down as a little list 3,4.

Â So 3,4 here is the instructions to do 3e1 hats plus 4e2 hats.

Â If you think about it, my choice of e1 hat and e2 hat here is kind of arbitrary.

Â It depends entirely on the way I set up the coordinates.

Â There's no reason I couldn't have set up some co ordinate system

Â at some angle to that, or you can use vectors to find the axis

Â that weren't even at 90 degrees to each other or were of different lengths.

Â I could still have described r as being some sum of some vectors I used to define

Â the space.

Â 2:32

And I've defined it in terms of the coordinates e.

Â And I could then describe r in terms of you, you're using, those vectors b1 and b2.

Â It's just the numbers in r would be different.

Â So, we call the vectors we used to define this space, these guys e or

Â these guys b, we call them basis vectors.

Â So the numbers I've used to define r only have any meaning when I know

Â about the basis vectors.

Â So r referred to these basis vectors e is 3,4.

Â But r referred to the basis vectors b also exists.

Â 3:12

We just don't know what the numbers are in there.

Â So this should be kind of amazing.

Â r, the vector r, has some existence in a deep sort of mathematical sense completely

Â independently of the coordinate system we use to describe, the numbers in

Â the list describing r.

Â r, the vector takes us from there,

Â from the origin to there still exists, independently of the numbers used in her.

Â Which is kind of neat, right?

Â Sort of fundamentally, sort of idea.

Â Now, if the new basis vectors, these guys b are at 90 degrees to each other,

Â then it turns out the projection product has a nice application.

Â We can use the projection or dot product to find out the numbers for

Â r in the new basis, b, so long as we know what the b's are in terms of e.

Â So here I've described b1 as being 2,1,

Â as being e1 plus e2 twice e1 plus 1 e2.

Â And I've described b2 as being minus 2e1's plus 4e2's.

Â And if I know b in terms of e, I'm going to be able to

Â use the projection product to find r described in terms of the b's.

Â But this is a big if, the b1 and b2 have to be at 90 degrees to each other.

Â If they're not, we end up being in big trouble and need matrices to do what's

Â called a transformation of axes, from the e to the b set of basis vectors.

Â We'll look at matrices later, but this will help us out a lot for now.

Â 4:59

I look down from here and project down at 90 degrees I get a length here for

Â scalar product, and that scalar projection is the shadow of r1 to b1.

Â And the number of the scalar projection describes how much of this vector I need.

Â And the vector projection is going to actually give me a vector

Â in the direction of b1 of length equal to that projection.

Â 5:26

Now if I take the vector protection of r onto b2 going this way, I'm

Â going to get a vector in the direction of b2 of length equal to that projection.

Â And if I do a vector sum of that

Â vector projection plus this guy's vector projection, I'll just get r.

Â So if I can do those two vector projections, and add up their

Â vector sum, I'll then have our b being the numbers in those two vector projections.

Â And so I found how to get from r in the e set of

Â basis vectors to the b set of basis vectors.

Â Now how do I check that these two new basis vectors are at 90 degrees to each

Â other?

Â Well, I just take the dot product.

Â So we said before the dot product cos theta

Â was equal to the dot of two vectors together,

Â so b1 and b2, divided by their lengths.

Â So if b1.b2 is 0, then cosine theta is 0, and

Â cosine theta is 0 if they're 90 degrees to each other, if they're orthogonal.

Â So I don't even need to calculate a thing, so I just calculate the dot product.

Â So b1.b2 here, I take 2 times minus 2 and

Â I add it to 1 times 4, which is, minus 4 plus 4 which is 0.

Â So these two vectors are at 90 degrees to each other.

Â So it's going to be safe to do the projection.

Â So having talked through it, let's now do it numerically.

Â 6:58

So if I want to know what r described in the basis e,

Â and r is pink right, if I take r in the basis e,

Â and I'm going to dot him with b1,

Â and the vector projection divides by the length of b1 squared.

Â So r in e dotted with b1 is going to be 3

Â times 2 plus 4 times 1, 4 times 1,

Â divided by the length of b1 squared.

Â So that's the sum of the squares of the components of b.

Â So that's 2 squared plus 1 squared.

Â So that gives me 6 plus 4 is 10, divided by 5 which is 2.

Â So this projection here is of length 2 times b1.

Â So that projection there,

Â that vector is going to be 2 times b1.

Â So that is in terms of the original set of vectors e,

Â r_e.b1 over b1 squared times b1 is 2 times

Â the vector 2,1, is the vector 4,2.

Â 9:32

Now if I add those two together,

Â 4,2 this bit, that vector projection,

Â plus this vector projection, so

Â this guy is going to be half b2 plus half -2,4 is -1,2.

Â If I add those together, I've got 3,4,

Â which is just my original vector r, 3,4 in the basis e.

Â So in the basis of b1 and b2, r_b is going to be 2

Â one-half, very nice, 2,1/2.

Â So actually in the basis b, it's going to be 2,1/2, there.

Â 10:51

So the basis vectors we use to describe the space of data, and

Â choosing them carefully to help us solve our problem,

Â will be a very important thing in intermediate algebra, and in general.

Â And what we've seen is we can move the numbers in the vector,

Â we used to describe a data item from one basis to another.

Â We can do that change just by taking the dot or projection product in

Â the case where the new basis factors are orthogonal to each other.

Â