Can someone explain to me what he is doing here, transformation matrices, weee

[mr_coffee]

Hello everyone, i posted 2 pictures, they are the same problem, just couldn't fit it.
http://img222.imageshack.us/img222/1227/lastscan5vx.jpg
http://img222.imageshack.us/img222/1681/lastscan29fm.jpg

I don't understand the majority of the steps, like hwy does he multiply
2*[1 2] - [2 1] = 3j?
also
he then says
j = 2/3[ 1 2] - 1/3[2 1] ?
this one really doesn't make sense:
i = [1 2] - 2(2/3*[1 2] - 1/3[2 1])
The examples in the book are easy and i understand then, they always had either i given to u or j given to you. like
T[ 1 0] = [2 -3] but this one doesn't give u anything so i know u have to manipulate the matrices to get the standard, i = [1 0] j = [0 1] but I'm confused on his logic behind it, can someone explain to me why he did what he did'? thanks!

 

[HallsofIvy]

Hello everyone, i posted 2 pictures, they are the same problem, just couldn't fit it.
http://img222.imageshack.us/img222/1227/lastscan5vx.jpg
http://img222.imageshack.us/img222/1681/lastscan29fm.jpg
I don't understand the majority of the steps, like hwy does he multiply
2*[1 2] - [2 1] = 3j?

He has already said that i= [1 0] and j= [0 1]. [a b]= [a 0]+ [0 b]= a[1 0]+ b[0 1]= ai+ bj. 2*[1 2]- [2 1]= [2 4]- [2 1]= [2- 2 4- 1]= [0 3] which is just another way of writing 0i+ 3j= 3j.

also
he then says
j = 2/3[ 1 2] - 1/3[2 1] ?

(2/3)[1 2]- (1/3)[2 1]= [2/3 4/3]- [2/3 1/3]= [2/3- 2/3 4/3- 1/3]=
[0 1]= 0i+ 0j

this one really doesn't make sense:
i = [1 2] - 2((2/3)[1 2]- (1/3)[2 1])

We've already established that (2/3)[1 2]- (1/3)[2 1]= [0 1] so
[1 2]- 2((2/3)[1 2]- (1/3)[2 1])= [1 2]- [0 2]= [1-0 2- 2]= [1 0] which is the same as i.

The examples in the book are easy and i understand then, they always had either i given to u or j given to you. like
T[ 1 0] = [2 -3] but this one doesn't give u anything so i know u have to manipulate the matrices to get the standard, i = [1 0] j = [0 1] but I'm confused on his logic behind it, can someone explain to me why he did what he did'? thanks!

One more time, i is just a short way of writing [1 0], j is just a short way of writing [0 1]. "ai+ bj" is a short way of writing [a b].

 

[mr_coffee]

Thank you so much Ivy, that made a lot more sense, I worked through it again and I'm stuck on this part, I understand all the steps to get to 3j; or [0 3] = 0i + 3j = 3j;
But then we go into another series of calculations to get it to [0 1] which is just j. Was the point of finding 3j first, so you could perform the following calculation?
2/3[1 2] - 1/3[2 1]? since u found 3j, u know u can get [0 1] by multplying by 1/3? so is that why u have the -1/3? And ur multpying 2/3 by [1 2] because ur trying to get a 0 on top and a 1 on the bottom right? so u get [0 1] which is ur j. Is my reasoning right? Thanks again!

I just added to this post, i think i get the top part, but this one really doesn't make sense:
he writes:
i = -1/3[1 2] + 2/3[2 1] = [1 0] yes i see that it equal i, but how does that relate to all the work we just did above to find i and j? Because we orginally wrote to find i, we had i = [1 2] - 2(2/3[1 2] - 1/3[2 1]) = [1 0] = i; but then he writes:
i = -1/3[1 2] + 2/3[2 1] which isn't what we had above?

 

[mr_coffee]

Wow, 2 hours of working through examples and i finally figured out what's going on with transformations! thanks for the help you were telling me the right way, it was just going in and my brain was like f it. :)