Function Composition of Multivariate Functions

[Prof. 27]

Homework Statement

This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of composition itself on Wikipedia (to partial composition I believe it says). Yet we have not learned this, nor is it anywhere to be found in the lecture notes, which until this point have been excellent. My question is, is there something I'm not understanding? Is this possible with traditional composition? I'll show the idea I had but I don't know if it is anywhere near correct. Bold parts are not currently relevant to what I don't understand.

Question:

Let T : R² → R³ be T(x, y) = (2x−3y, 4x−11y, x), and let S : R³ → R² be S(u, v, w) = (u + v − w, 2u − 3v + 5w). i) Calculate T ◦ S; ii) Write the matrices of T and S; iii) Find the matrix of T ◦ S and relate it to the matrices in ii).

My Attempt:
We know at least that ToS must be in R⁵, because the Cartesian product of two sets of finite dimensions equals the dimensions added.

T(x,y) = 2x-3y, 4x-11y, x
S(u,v,z) = u+v-w, 2u-3v+5w

a = u+v-w
b = 2u-3v+5w

T(S(u,v,z) = T(a,b) = 2a-3b, 4a-11b, a

The problem: Isn't assigning a,b to x,y respectively arbitrary? Couldn't I just as well say,

T(S(u,v,z) = T(b,a) = 2b-3a, 4b-11a, b
Since a does not equal b, T(b,a) does not equal T(a,b) (at least usually).

Help?

Homework Equations

None

The Attempt at a Solution

Google
Stack Exchange
Lecture Notes
Textbook, Calculus by Michael Spivak

 

[SammyS]

Homework Statement

This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of composition itself on Wikipedia (to partial composition I believe it says). Yet we have not learned this, nor is it anywhere to be found in the lecture notes, which until this point have been excellent. My question is, is there something I'm not understanding? Is this possible with traditional composition? I'll show the idea I had but I don't know if it is anywhere near correct. Bold parts are not currently relevant to what I don't understand.

Question:

Let T : R² → R³ be T(x, y) = (2x−3y, 4x−11y, x), and let S : R³ → R² be S(u, v, w) = (u + v − w, 2u − 3v + 5w). i) Calculate T ◦ S; ii) Write the matrices of T and S; iii) Find the matrix of T ◦ S and relate it to the matrices in ii).

My Attempt:
We know at least that ToS must be in R⁵, because the Cartesian product of two sets of finite dimensions equals the dimensions added.

T(x,y) = 2x-3y, 4x-11y, x
S(u,v,z) = u+v-w, 2u-3v+5w

a = u+v-w
b = 2u-3v+5w

T(S(u,v,z) = T(a,b) = 2a-3b, 4a-11b, a

The problem: Isn't assigning a,b to x,y respectively arbitrary? Couldn't I just as well say,

T(S(u,v,z) = T(b,a) = 2b-3a, 4b-11a, b
Since a does not equal b, T(b,a) does not equal T(a,b) (at least usually).

Help?

Homework Equations

None

The Attempt at a Solution

Google
Stack Exchange
Lecture Notes
Textbook, Calculus by Michael Spivak

When using the terminology:: "Ordered Pair", and/or "Ordered Triple";

the word order is very important.

 

[Prof. 27]

Thanks, that helps a lot.