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Proposition 1.3. Let g: R → R be a continuous function. Consider q P R given by (f) (g(x)f(x)dx, where ƒ is a polynomial. Then op is a linear functional. Project Problem 3: Prove the above Proposition 1.3. =

Proposition 1.3. Let g: R → R be a continuous function. Consider q P R given by (f) (g(x)f(x)dx, where ƒ is a polynomial. Then op is a linear functional. Project Problem 3: Prove the above Proposition 1.3. =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1 Linear Functionals Definition 1.1. Let V be a vector space over R. A linear functional on V is a linear map q: V → R. Examples: Example 1) Consider V = R", and q: R" → R given by q (0) = x1 + + xn, where = (x1,x2,...,xn). Then is a linear functional. 9 The next proposition provides a more general example of a linear functional on R". Proposition 1.2. Let a₁, a2,,an be fixed real numbers. Let q: R" → R be given by p() = α₁x1 + x2x2 + ... + αnxn, where = (x₁,x2,...,xn). Then q is a linear functional. Project Problem 1: Prove the above Proposition 1.2. Example 2) Consider V = M₂ (R), where M₂ (R) denotes the set of all 2 × 2 matrices with real entries. Let Tr: M₂ (R) → R be given by Tr ) = a + d. Then Tr is a linear functional on M2 (R), called the Trace. More generally, we can define a similar linear functional on M,, (R), where M,, (R) denotes the set of all n x n matrices with real entries. Let Trn Mn (R)→ R be given by Trn(X) = x1,1 + x2,2+ + xnn. Here X is an n x n matrix, and xi,i denotes the ith diagonal entry of X. Then Trn is a linear functional. Project Problem 2: Prove that Trn: M₂ (R) → R is a linear functional. Example 3) Consider V = P, the set of all polynomials with real coefficients. A linear functional on P is given by integration, (f) = f(x)dx, where ƒ is a polynomial. A more general form of a linear functional on is given in the next proposition. Proposition 1.3. Let g R → R be a continuous function. Consider q P → R given by q(f) g(x)ƒ(x)dx, where ƒ is a polynomial. Then qp is a linear functional. = Project Problem 3: Prove the above Proposition 1.3. In the next section, we shall describe all linear functionals on V, where V is a finite dimensional vector space over R.
Transcribed Image Text:1 Linear Functionals Definition 1.1. Let V be a vector space over R. A linear functional on V is a linear map q: V → R. Examples: Example 1) Consider V = R", and q: R" → R given by q (0) = x1 + + xn, where = (x1,x2,...,xn). Then is a linear functional. 9 The next proposition provides a more general example of a linear functional on R". Proposition 1.2. Let a₁, a2,,an be fixed real numbers. Let q: R" → R be given by p() = α₁x1 + x2x2 + ... + αnxn, where = (x₁,x2,...,xn). Then q is a linear functional. Project Problem 1: Prove the above Proposition 1.2. Example 2) Consider V = M₂ (R), where M₂ (R) denotes the set of all 2 × 2 matrices with real entries. Let Tr: M₂ (R) → R be given by Tr ) = a + d. Then Tr is a linear functional on M2 (R), called the Trace. More generally, we can define a similar linear functional on M,, (R), where M,, (R) denotes the set of all n x n matrices with real entries. Let Trn Mn (R)→ R be given by Trn(X) = x1,1 + x2,2+ + xnn. Here X is an n x n matrix, and xi,i denotes the ith diagonal entry of X. Then Trn is a linear functional. Project Problem 2: Prove that Trn: M₂ (R) → R is a linear functional. Example 3) Consider V = P, the set of all polynomials with real coefficients. A linear functional on P is given by integration, (f) = f(x)dx, where ƒ is a polynomial. A more general form of a linear functional on is given in the next proposition. Proposition 1.3. Let g R → R be a continuous function. Consider q P → R given by q(f) g(x)ƒ(x)dx, where ƒ is a polynomial. Then qp is a linear functional. = Project Problem 3: Prove the above Proposition 1.3. In the next section, we shall describe all linear functionals on V, where V is a finite dimensional vector space over R.
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