# Thread: Linear maping, maybe?

1. ## Linear maping, maybe?

Let $\displaystyle S: R^3 -> R^2$ and $\displaystyle T:R^2 -> R^3$ be defined by

$\displaystyle S(x_1, x_2, x_3) = (x_2 + x_3, x_1)$, $\displaystyle T(x_1, x_2) = (x_2, x_1, x_1 + x_2)$

Find expressions for ST and TS.

Any help please?

2. Hello,

Originally Posted by Deadstar
Let $\displaystyle S: R^3 -> R^2$ and $\displaystyle T:R^2 -> R^3$ be defined by

$\displaystyle S(x_1, x_2, x_3) = (x_2 + x_3, x_1)$, $\displaystyle T(x_1, x_2) = (x_2, x_1, x_1 + x_2)$

Find expressions for ST and TS.

Any help please?
Are you looking for the composition of S with T and T with S ? o.O

Let's see for $\displaystyle SoT(x_1,x_2)$ :

$\displaystyle T(x_1,x_2)=(x_2, x_1, x_1 + x_2)$

---> $\displaystyle SoT(x_1,x_2)=S(T(x_1,x_2))=S(x_2, x_1, x_1 + x_2)$

We know that $\displaystyle S(m,n,p)=({\color{red}n+p},{\color{blue}m})$.
I renamed this on purpose, because it would confuse you.
Here :
$\displaystyle m=x_2$ (1)
$\displaystyle n=x_1$ (2)
$\displaystyle p=x_1+x_2$ (3)

Therefore $\displaystyle {\color{red}n+p}=(2)+(3)=\boxed{2x_1+x_2}$
And $\displaystyle \boxed{{\color{blue}m}=x_2}$

Hence :

$\displaystyle S(x_2, x_1, x_1 + x_2)=(2x_1+x_2, \ x_2)$

---> $\displaystyle \boxed{SoT(x_1, \ x_2)=(2x_1+x_2, \ x_2)}$

Is it what you wanted ? Does it help ?

3. Yeah i think i get it now.

I dont know how to explain it exactly...

Is this right...
Since $\displaystyle m = x_2$ (1) for example...

Any $\displaystyle x_1$ in the equation $\displaystyle (x_2 + x_3, x_1)$ must be replaced by $\displaystyle m = x_2$...

Dunno if that sounds right the way ive explained it but it works when i try it cos i got $\displaystyle TS = (x_1, x_2 + x_3, x_1 + x_2 + x_3)$ which was the right answer!

4. Hmm how to explain... Actually, I replace by m, n and p because it was redundant and really confusing.
It's equivalent to say "the first coordinate of the image of a triplet by T is the sum of its two last coordinates".

Originally Posted by Deadstar

Any $\displaystyle x_1$ in the equation $\displaystyle (x_2 + x_3, x_1)$ must be replaced by $\displaystyle m = x_2$...
It sounds strange to me... Because once you replace with m, n or p, you don't have to say "I'll replace it by...".

We know that $\displaystyle S(m,n,p)=({\color{red}n+p},{\color{blue}m})$.
I renamed this on purpose, because it would confuse you.
Here :
$\displaystyle m=x_2$ (1)
$\displaystyle n=x_1$ (2)
$\displaystyle p=x_1+x_2$ (3)

Therefore $\displaystyle {\color{red}n+p}=(2)+(3)=\boxed{2x_1+x_2}$
And $\displaystyle \boxed{{\color{blue}m}=x_2}$
This part was more for explaining the thing to you than for writing the correct answer.
You have your own way to explain it. I don't think your teacher would bother that much if you showed it your way. But make it understandable