Compositions of Linear Transformations

[Dgray101]

Homework Statement

(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]

Homework Equations

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?

 

[Mark44]

Homework Statement

(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]

Homework Equations

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?

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.