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I don't quite understand. Could you please explain?

On it.

Could you tell me what for, please?

it is a way of finding the answer...it will give you what
T(1,0,0)
T(0,1,0) and T(0,0,1) are

then T(x,y,z)=xT(1,0,0)+yT(0,1,0)+zT(0,0,1)

Is this correct |dw:1379741449340:dw| ?

no

Ok. And then what should I do?

then
T(1,0,0)=(1,0)
T(0,1,0)=(1,1)
T(0,0,1)=(0,1)

T(x,y,z)=xT(1,0,0)+yT(0,1,0)+zT(0,0,1)
=x(1,0)+y(1,1)+z(0,1)=(x+y,y+z)

Could you explain this with words please? I am not following you.

if this is confusing you can always try blockcolder's method...that will work too.

I just don't understand this at all. I want too, I just can't seem to get it in my head.

do you understand where my original matrix came from?

No, not really. I would really appreciate it if you could explain it step-by-step, please.

look at the first row...it is just
T(1,1,2)=(2,3)
1 1 2 | 2 3

Ok.

same with the 2nd and 3rd rows

then you row reduce it....this gives us what the standard basis vectors map to

And then what do we do?

then we use the linearity of T to get the answer

we want T(x,y,z)

T[(x,y,z)]=T[(x,0,0)+(0,y,0)+(0,0,z)]
=T(x,0,0)+T(0,y,0)+T(0,0,z)
=xT(1,0,0)+yT(0,1,0)+zT(0,0,1)

and then?

but where do you get (x,0),(y,y) and (0,z)?

T(1,0,0)=(1,0)
first row of matrix
1 0 0 | 1 0

oh, got it!

good

yes

Ok! By any chance, do you have practice problems or do you know where I could find any?

none typed up

you have a text book?

I do, but it doesn't have linear transformations in it.

what book do you have?

ok...I have a lot of linear algebra book...but I don't think I have that one

*books

I have like 10 of them...i'd have to check my office ;)

Its ok. My book is in Spanish anyway.
How much longer are you going to stay around?

not much longer...it is pretty late for me

2am

Ok! Thank you so much for your help, patience and time!

no problem...good luck on your exam!

Thank you!