Matrix Representation of Differentiation Operator for Subspace S in C[a,b]

In summary, the conversation is about finding the matrix representing the differentiation operator of a subspace in C[a,b] spanned by e^x, xe^x, and (x^2)e^x. The person is asking for help and the other person suggests trying to use the differential operator on each of the bases and rewriting the results as a linear combination. A hint is also given to consider the relevance of the coordinate vectors of the results.
  • #1
electricalcoolness
18
0
Another problem I can't figure out how to start.

Let S be the subspace of C[a,b] spanned by e^x , xe^x , (x^2)e^x . Let D be the differentiation operator of S. Find the matrix representing D with respect to [e^x, xe^x, (x^2)e^x ]
 
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  • #2
Well, what have you tried? Do you know how to find a matrix representing a linear transformation?
 
  • #3
Hint

Hint: Try using the differntial operator on each of your bases and then see if you can rewrite each of the results as a linear combination of the bases. After that you may wish to consider the relevance of the coordinate vectors of your results.
 

FAQ: Matrix Representation of Differentiation Operator for Subspace S in C[a,b]

What is a differentiation operator?

A differentiation operator is a mathematical operator that takes a function as input and produces its derivative as output. It is denoted by d/dx or ∂/∂x and is used to represent the rate of change of a function with respect to its independent variable.

What is a subspace S in C[a,b]?

A subspace S in C[a,b] is a subset of the space of continuous functions defined on the interval [a,b]. It satisfies the properties of a vector space, such as closure under addition and scalar multiplication, and contains the zero function.

How is the differentiation operator represented in a matrix form?

The differentiation operator for a subspace S in C[a,b] is represented by a matrix with the coefficients of the derivatives of the basis functions of S as its elements. The basis functions are chosen such that they form a spanning set for S, and the matrix has the dimensions of the number of basis functions in S.

What are the advantages of using matrix representation for the differentiation operator?

Using matrix representation for the differentiation operator allows us to perform computations and manipulations on the operator using matrix operations, which are well-studied and understood. This can simplify complex calculations and make it easier to apply the operator to functions within the subspace S.

How is the matrix representation of the differentiation operator used in practical applications?

The matrix representation of the differentiation operator is used in various fields of science and engineering, such as signal processing, control systems, and differential equations. It allows for efficient and accurate computation of derivatives, which are essential in modeling and analyzing real-world systems.

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