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.

 

[blue_raver22]

First of all, I understand your frustration with the material and the presentation of it. Linear algebra can be a challenging subject, especially when the material is not presented in a clear and organized manner.

To address your specific problem, let's start by breaking it down into smaller parts. First, we have a linear transformation T that takes a function f(t) and returns the definite integral of that function from 3 to -9. This means that T is a function that maps one function to another function.

Next, we have to find the matrix A of this linear transformation. This means that we need to find a way to represent this transformation using a matrix. In order to do this, we need to define a basis for both the vector space of polynomials of degree 3 (P_3) and the vector space of real numbers (R). A basis is a set of vectors that can be used to represent any vector in a given vector space. In this case, we can use the standard basis for both P_3 and R, which consists of the vectors (1, x, x^2, x^3) for P_3 and (1) for R.

Now, to find the matrix A, we need to find the values of T for each of the basis vectors in P_3. This will give us the columns of our matrix. For example, when we apply T to the basis vector 1, we get the definite integral of 1 from 3 to -9, which is -6. So, the first column of our matrix A will be (-6, 0, 0, 0).

Similarly, when we apply T to the basis vector x, we get the definite integral of x from 3 to -9, which is -36. So, the second column of our matrix A will be (-36, 0, 0, 0).

We can continue this process for the remaining basis vectors (x^2 and x^3) to get the full matrix A.

In summary, to find the matrix of a linear transformation involving integrals, we need to define a basis for the vector space and then apply the transformation to each basis vector to get the columns of our matrix. I hope this helps and good luck with your homework problems!