Homework Help: Compositions of Linear Transformations

1. Nov 2, 2013

Dgray101

1. The problem statement, all variables and given/known data

(ii) S ◦ T will be a linear transformation from P4 to R2. Write a formula for the value S(T (a4t4 + a3t3 + a2t2 + a1t + a0)) using the given formulas for T,S and use this to compute the matrix [S ◦T]B′′,B. (10p)

B'' = {e1 e2}
B' = {t4, t3, t2, t,1}

T: P4--> M2x2
T(a4t4 + a3t3 + a2t2 + a1t + a0) = ( (a0 +a4 +2a2) (-a1 + a3 - a2) )
( (a1+a3+a2) (a0-a4) )

S:M2x2 ---> R2
S( x1A1 + x2A2 +x3A3 + x4A4 ) = (x1 +x2)
(x3-x4)

Where A1=[1 0 ,0 -1] A2= [ 1 0, 0 1] A3= [0 1, -1 0] A4 = [0 1, 1 0]

2. Relevant equations

3. The attempt at a solution

I don't quite understand how we can get the linear transformation S(T) so be in the desired form. Because we get S ( 2x2 matrix) but the definition of S is not this?

Last edited by a moderator: Nov 2, 2013
2. Nov 2, 2013

Staff: Mentor

If you are sure you have copied the problem correctly, then there is a problem in how it is stated. S ° T makes no sense, but T ° S does make sense. Maybe that's what they're really asking for.

BTW, at the very least use ^ to indicate exponents. Instead of writing a4t4 + a3t3 + a2t2 + a1t + a0, you can write this: a4t^4 + a3t^3 + a2t^2 + a1t + a0.

Even better, click the Go Advanced button below the text entry area. This opens an advanced menu across the top. Use the X2 button to create exponents, and the X2 button to create subscripts.

Here is your polynomial with subscripts and exponents: a4t4 + a3t3 + a2t2 + a1t + a0. It takes a little extra time, but makes what you right much more readable.