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Now let's do that slowly.
How do you choose coordinate system?
And what is it that you do when you choose coordinate system?
Well of course you choose the origin first, right?
And then what?
And then you choose direction of the x-axis.
Right? So how you choose that?
Do you really choose arbitrarily?
Or you use some preferences coming from the problem?
Student: [...].
All right. You [...] identify the easiest direction, right?
Sometimes it is horizontal.
And sometimes it is not.
Right? So you identify easiest direction.
And you put your x-axis through that direction.
That's how you do it, right? That's the first step.
Now if I want to put a coordinate system in R^4, in that space ...
This is all I have to do.
I have to identify what are the easiest directions.
Now how would I choose easiest direction?
Well algebraically I can choose any quadruple and plane well that direction is the easiest.
Because I don't see any difference between those.
But ...
But I have some preferred directions.
I really have ...
... an interesting direction here.
Don't you want to put the first coordinate axis ...
... along the direction of the matrix one, zero, zero, one?
Because that's a good matrix. That's the matrix any geometer of physicist could understand easily.
So basically relation to physics and geometry suggests ...
... the choice of coordinate system.
Suggests that some way is easier than others.
That's all it does at this point.
Easier preference.
Otherwise you can choose anything.
But ... So let's choose this one.
And then how do you continue ...
... introducing coordinate system in engineering?
Pick y-coordinate. How do you pick it?
Perpendicular, right?
So if you choose x to be in this direction you want y ...
... coordinate perpendicular. Why is that?
Why perpendicular?
Why don't you choose it ...
... at some angle?
So let's keep that question in mind.
There will be an important answer coming later ...
... from algebra. So ...
So we want it to be perpendicular.
OK.
Now do I have another preferred matrix?
Well ...
Zero, minus one, one, zero could be another one, right?
So I may want to choose another axis in direction of zero, minus one, one, zero.
Now yes, I can. But is it perpendicular to that direction?
Now what does it mean to be perpendicular?
Dot product equals zero.
Right?
Because we know exactly the algebraic way of checking if two vectors are perpendicular.
We don't have to draw any picture here.
And actually we cannot. Right?
We cannot visualize four-dimensional space.
But algebraically we can look at this matrix zero, minus one, one, zero.
Thought of it ...
Think of it as a vector.
We can think of that as a vector.
One, zero, zero, one.
And we can think of dot product.
So [...] the dot product is? How to compute the dot product?
Remember how we computed dot product for two-dimensional vectors.
And three-dimensional vectors.
Can you compute it for four-dimensional vectors now?
All right.
Well people in the first rows can.
What about those people in the last rows?
Can you?
So zero times one, right?
So zero times one plus negative one times zero ...
... plus one times zero.
So you multiply the corresponding values.
Plus zero times one.
You add the products together.
And in this case you get zero.
Well in general you don't but in this case you do.
And that tells us exactly that these two vectors are perpendicular indeed.
And that is indeed a great choice ...
... for the coordinate system.
And we're inspired by the process.
Well you see we are doing purely geometric thinking, right?
But what we do -- the action -- is algebraic.
Right? Isn't that great?
To think one way and do something else.
So ...
Er ... So what would be the next thing to choose?
Because how many more do you want?
Two more, right? Two-dimensional space, we have two so far, we need two more.
Now is that a good direction?
Direction in the ...
Direction of zero, zero, zero, zero. It's not even a direction, right?
So that doesn't suggest anything.
What about this one?
Well the thing is that this matrix ...
... is equal ...
... as we computed last time.
So this matrix cosine alpha, negative sine alpha ...
... sine alpha and cosine alpha ...
... is equal ... Well think of it as a vector.
Right?
Now this vector is equal to ...
... cosine alpha times the matrix one, zero, zero, one ...
... plus sine alpha ...
... times another vector zero, minus one, one, zero.
Now what does that mean?
From our point of view what does it mean ...
... from this four-dimensional geometry point of view?
It means that if you take this vector ...
... in that four-dimensional space ...
... and you make that vector in that four-dimensional space ...
... and if you multiply the first by some number ...
... which amounts to what?
Making it [...] shorter, right?
Well in this case it's probably shorter.
So you multiply that by some number.
And you multiply this by some number making it in this case shorter also.
And then you add those together.
And you know how to add those two together, right?
Then the vector you get is what?
It's still in the same plane.
We are not gaining any four-dimensional insight ...
... by bringing any of those vectors.
All these belong to the same plane in four-dimensional space.
So it doesn't help to use any of those matrices.
And ...
The way we decided about that is purely algebraically.
So what should we make?
Because I run out of examples.
I'm stuck.
So what should we ...
The other two matrices I would want to use.
Let's choose the third one.
This is number one, that is number two.
What's number three?
Now what do we expect from that third vector?
We expect it to be ...
... perpendicular to both of those, right?
Now can you think of a matrix ...
... so that you multiply -- you dot product with this and get zero ...
... and you dot product with that and get zero?
Well let's ...
... [...] this one.
Dot product with this ...
... is obviously zero, right?
And dot product with that is what?
Minus one times one ...
... and one times one. Those cancel.
And the dot product is zero.
So that matrix is perpendicular to both.
And it's so simple, isn't it?
It's of the same simplicity as the other two.
Now can you suggest the fourth one?
Negative one, zero, zero, one.
Something like that, right?
So the dot product of this with that or that is zero obviously.
Right? Because any time you multiply corresponding entries you get zero.
And dot product of this and that is zero because minus one plus one cancel also.
And now I get fourth. Do I have enough?
Yes, I do have enough.
Now what is our expectation about that achievement?
Well the expectation is that ...
... now we should be able to express any matrix ...
... in terms of those four.
Only.
Right? Can we?
What should be a way?