Another Linear Transformation problem

[mlarson9000]

Homework Statement

Let F be the vector space of all functions mapping R into R, and letT:F-F be a linear transformationsuch that T(e^2x)=x^2, T(e^3x)= sinx, and T(1)= cos5x. Find the following, if it is determined by this data.

Homework Equations

a. T(e^5x)
b. T(3+5e^3x)
c. T(3e^4x)
d. T((e^4x + 2e^5x)/e^2x)

The Attempt at a Solution

a. T(e^2x)*T(e^3x)= (x^2)sinX?
b. 3T(1)+5T(e^3x)=3cosx + 5sinx
c. 3T(e^2x)T(e^2x)= 3x^4
d. T((e^4x)/(e^2x))+2T((e^5x)/(e^2x))= T(e^2x)+2T(e^3x)= (x^2) + (2sinX)

Is this right?

 

[Mark44]

Looks OK. I didn't check the last one very closely, but you have the right idea.

 

[HallsofIvy]

Homework Statement

Let F be the vector space of all functions mapping R into R, and letT:F-F be a linear transformationsuch that T(e^2x)=x^2, T(e^3x)= sinx, and T(1)= cos5x. Find the following, if it is determined by this data.

Homework Equations

a. T(e^5x)
b. T(3+5e^3x)
c. T(3e^4x)
d. T((e^4x + 2e^5x)/e^2x)

The Attempt at a Solution

a. T(e^2x)*T(e^3x)= (x^2)sinX?
b. 3T(1)+5T(e^3x)=3cosx + 5sinx
c. 3T(e^2x)T(e^2x)= 3x^4
d. T((e^4x)/(e^2x))+2T((e^5x)/(e^2x))= T(e^2x)+2T(e^3x)= (x^2) + (2sinX)

Is this right?

I am very hesitant to disagree with Mark44, but generally it is NOT true that T(uv)= T(u)T(v) for a vector space- in fact, the product of two vectors is not part of the definition of "vector space". Is the product of functions somehow being used as the "vector sum"? If so what is the "negative" of the 0 function?

 

[g_edgar]

Solutions a. and c. are incorrect, for the reason cited by HallsOfIvy.

"linear transformation" does not specify what happens on products.

 

[Mark44]

Mea culpa

 

[mlarson9000]

So are any of these solveable other than b. based on the given information?

 

[Mark44]

Parts b and d can be done with the information given; parts a and c cannot. Your answer for b is partly correct (T(1) = cos(5x), not cos(x)), and your answer for d is correct.