Linear Transformations (T o S?)

[dvvv]

Homework Statement

Let T : R2 -> R2 be the linear transformation defined by the formula
T(x, y) = (2x + 3y,−x − y).

Let S : R2 -> R2 be the linear transformation whose matrix is
3 −1
2 4

i. Write down the matrix of T.
ii. Calculate the matrices of the linear transformations T o S and S o T, indicating which is which.
iii. Calculate the image of the line L whose equation is x + y = 2, under the transformation S o T.

Homework Equations

The Attempt at a Solution

i. I said
2 3
-1 -1

Then for the next parts I don't know how you get ToS or SoT?

 

[praharmitra]

Can you tell us what the definition for ToS and SoT ?

 

[dvvv]

Well I can't seem to find any definition, but if you had ToS(u), that would be equal to T(S(u)). Similarly SoT(u) would be S(T(u)) I assume.

 

[praharmitra]

Thats right! Now, you agree that action of T (or S) on any (x,y) can be written as

$$T(x,y) = \left[ \begin{array}{cc} 2 & 3 \\ -1 & -1 \end{array} \right] \left[\begin{array}{cc} x \\ y \end{array} \right]$$

One can write a similar action of S. Action of T on (x,y) gives another vector (x',y'). So SoT(x,y) would mean S(T(x,y)) = S(x',y') = (x'', y'').

Now can you write the matrix for SoT?

 

[dvvv]

I got
12 10
-5 -3

I just multiplied T by S. I wasn't sure what to do since it's just ToS not ToS(x,y) or something like that.

 

[praharmitra]

Yes, that's ToS. I hope that's what you meant to calculate?

 

[dvvv]

Oh well I thought I was getting SoT but I got confused apparently. I get it now. SoT is
7 4
0 -2
edit: made a mistake, I'll fix it in a second.

 

[praharmitra]

Kindly check the calculation. The answer is
7 10
0 2

 

[dvvv]

Yeah sorry about that.

 

[dvvv]

For the last part I did
$$SoT(x,y) = \left[ \begin{array}{cc} 7 & 10 \\ 0 & 2 \end{array} \right] \left[\begin{array}{cc} x \\ y \end{array} \right]$$
and I got (7x+10y, 2y). I substituted that into x+y=2 and I got 7x + 12y = 2.
Is that right?

 

[praharmitra]

No. See, after applying, you move to a new space (x',y'). Your equations give you

x' = 7x + 10y
y' = 2y

What you are required to do is to find the relation b/w x' and y' using that between x and y.

 

[dvvv]

x = (-2x' + y')/7
y = 3x' + y'

I substituted that into x+y=2 and I got
19x' + 8y' = 14

edit: made a mistake,..one sec

 

[dvvv]

x = (2x' - y')/14
y = y'/2

Sub'd in and got x' + 3y' = 14

 

[praharmitra]

Please check your algebra - I got 2x'-3y' = 28

 

[dvvv]

OK. I keep making stupid mistakes. Thanks for all your help.