Compositions of Linear Transformations

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  • #1
Dgray101
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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?
 
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  • #2
Dgray101 said:

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.
 

FAQ: Compositions of Linear Transformations

1. What is a composition of linear transformations?

A composition of linear transformations is a mathematical operation that combines two or more linear transformations to create a new transformation. It involves applying one transformation followed by another, resulting in a single transformation that is the combination of the individual transformations.

2. How do you represent a composition of linear transformations?

A composition of linear transformations can be represented using matrix multiplication. The matrices representing the individual transformations are multiplied together in the order in which they are applied. The resulting matrix represents the composition of the transformations.

3. What is the order of operations in a composition of linear transformations?

In a composition of linear transformations, the order of operations is from right to left. This means that the last transformation in the composition is applied first, followed by the second to last, and so on until the first transformation is applied. This is the opposite of the order of operations in regular algebraic expressions.

4. Can a composition of linear transformations be reversed?

Yes, a composition of linear transformations can be reversed by finding the inverse of each individual transformation and applying them in reverse order. This will result in the original transformations being applied in reverse order, effectively reversing the composition.

5. What are some real-world applications of compositions of linear transformations?

Compositions of linear transformations are used in various fields of science and engineering, including computer graphics, physics, and robotics. They are also used in data analysis and machine learning to transform and manipulate data. In everyday life, they can be seen in the transformation of geometric shapes and in the creation of 3D animations.

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