Matrix of a linear transformation for an integral?

[marathon]

i am having trouble with some homework problems in my linear algebra course... the book is brescher and the teacher is sort of a rambling nutcase whose presentation of material is anything but 'linear', and very difficult for me to follow. similarly the book contains problems that i can't seem to suss out based on the chapter. this is for the homework for a section on 'introduction to linear spaces' which mostly gave examples of 'how to tell if this is a subspace of this'...

the problem is: find the matrix A (a vector with four components) of the linear transformation T(f(t)) = the definite integral of f(t) from 3 to -9 with respect to the standard bases for P_3 and R. there is nothing in the chapter about integrals as matrices, and what does it mean to have two bases at once...?? thanks

 

[HallsofIvy]

Is there anything in your book about itegrals as linear transformations? Or about writing a general linear transformation as a matrix? Those are what you need here.

An integral is a linear transformation: \int af(x)+ bg(x)dx= a\int f(x)dx+ b\int g(x)dx.

To write a linear transformation from vector space U to vector space V, given ordered bases for each, do the following. Apply the linear transformation to the first vector in the ordered basis for U. That will be in V so can be written as a linear combination of the ordered basis for V. The coefficients of that linear combination will be the first column in the matrix. Do the same with the second vector in the ordered basis for U to get the second column, etc.

 

[Fredrik]

the problem is: find the matrix A (a vector with four components) of the linear transformation T(f(t)) = the definite integral of f(t) from 3 to -9 with respect to the standard bases for P_3 and R. there is nothing in the chapter about integrals as matrices, and what does it mean to have two bases at once...?? thanks

The relationship between linear operators and matrices is explained e.g. in post #3 in this thread. (Ignore the quote and the stuff below it).

 

[marathon]

oh i see; yeah i finally found the appropriate section - two chapters ahead..! thanks.