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

the coordinates in which all of our vectors exist.

But it turns out in doing this thing of projecting,

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

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

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

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

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

from the origin to some point in space.

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

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

What we haven't talked about so

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

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

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

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

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

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

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

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

e5 hat, e 1 million hat, whatever.

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

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

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

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

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

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

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

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

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

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

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

the space.

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

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

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

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

these guys b, we call them basis vectors.

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

about the basis vectors.

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

But r referred to the basis vectors b also exists.

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

So this should be kind of amazing.

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

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

the list describing r.

r, the vector takes us from there,

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

Which is kind of neat, right?

Sort of fundamentally, sort of idea.

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

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

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

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

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

as being e1 plus e2 twice e1 plus 1 e2.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

And if I do a vector sum of that

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

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

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

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

basis vectors to the b set of basis vectors.

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

other?

Well, I just take the dot product.

So we said before the dot product cos theta

was equal to the dot of two vectors together,

so b1 and b2, divided by their lengths.

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

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

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

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

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

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

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

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

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

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

and I'm going to dot him with b1,

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

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

times 2 plus 4 times 1, 4 times 1,

divided by the length of b1 squared.

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

So that's 2 squared plus 1 squared.

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

So this projection here is of length 2 times b1.

So that projection there,

that vector is going to be 2 times b1.

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

r_e.b1 over b1 squared times b1 is 2 times

the vector 2,1, is the vector 4,2.

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Now if I add those two together,

4,2 this bit, that vector projection,

plus this vector projection, so

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

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

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

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

one-half, very nice, 2,1/2.

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

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

choosing them carefully to help us solve our problem,

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

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

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

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

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