Now, let's look at how we do the transformation of a vector that's already in a change basis. Last time, we looked at bears basis. Bears basis had a first axis of three, one and a second axis of one, one. Let's say I have a vector x, y defined in bears basis. As I want to transform it by doing something like a rotation of 45 degrees. But the problem is, I don't know how to write a 45 degree rotation in bears funny coordinate system. I only know how to write down a 45 degree rotation in my normal 1, 0, 0, 1 system. So in my system which is 1, 0, 0, 1 a 45 degree rotation rotates 1, 0 up like that. So if it's still a unit vector at 1, 0 over root 2, 1, over root 2, that is a normalized 1, 1. It takes 0, 1 round to minus 1 over root 2, 1 over root 2. That is that 45 degrees there, that 45 degrees there. So I can write it down the rotation in my notation. Let's call it R being a 45 degree rotation, as being 1 over root 2 times 1, 1 minus 1, 1. That's what a 45 degree rotation is in my world. So what I need to do is first transform the vector x, y into my basis. I do that by multiplying it by B. Then I can apply my nice sensible rotation R to that vector that's now in my basis. So what I get here when I do B, I've got the vector in my coordinate frame. Now the problem is bear doesn't care about my world, he wants to get the rotation in his basis. So then I have to transform the resulting vector, back into bears basis. I do that by applying B to the minus 1. B to the minus 1, I get by flipping the terms on the leading diagonal, taking minus the off-diagonal terms, and dividing by the determinant here which is 2 so I multiply it by a half. So that then gives me, the vector back in bears frame. So overall what I've done, is I've done B to the minus 1 times R, times B. What that's giving me, is it's given me the rotation, in bears coordinate system. Which is really neat. So now what we have to do is do the sums. When we do that R, B gives us this, and B to the -1 RB gives us this. I've written them down there, so pause and ponder if you want to verify those on your own. So this is what a 45 degree rotation looks like in bears coordinate system. Notice that it's completely different to the one in my standard basis. It isn't very easy necessarily or obvious to intuit just out of your head, you have to do the calculation. So if you want to do some translation but in some funny basis, this equation B to the minus 1 RB, is going to be very useful. To step back here, the point is that if we want to transform to normal form normal coordinate systems, then the translation matrices also change. We have to be mindful of that. This is the algebra you see all the time. We've got the transformation matrix R, wrapped around by B, B is the minus 1, that does the translation from my world to the world of the new basis system. So thanks a lot bear, you really helped us out to understand all this stuff. So what we've done in these two videos is we've looked at how the numbers in a vector change, where we change the basis. We thought about the coordinate systems and how to do transformations in non-orthogonal coordinate systems. It's been quite hard work, but this really sets us up for example in principal component analysis, to operate in different basis systems.