Project Problem 2: Prove that Trn: M₁, (R) → R is a linear functional. Example 3) Consider V = P, the set of all polynomials with real coefficients. A linear functional on 1 P is given by integration, p(f) = f(x)dx, where ƒ is a polynomial. A more general form of a linear 0 functional on is given in the next proposition. Proposition 1.3. Let g R → R be a continuous function. Consider P → R given by q(f) =
Project Problem 2: Prove that Trn: M₁, (R) → R is a linear functional. Example 3) Consider V = P, the set of all polynomials with real coefficients. A linear functional on 1 P is given by integration, p(f) = f(x)dx, where ƒ is a polynomial. A more general form of a linear 0 functional on is given in the next proposition. Proposition 1.3. Let g R → R be a continuous function. Consider P → R given by q(f) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please solve problem 2
![# Linear Functionals
## Definition 1.1
Let \( V \) be a vector space over \( \mathbb{R} \). A **linear functional** on \( V \) is a linear map \( \varphi : V \to \mathbb{R} \).
### Examples:
**Example 1**: Consider \( V = \mathbb{R}^n \), and \( \varphi : \mathbb{R}^n \to \mathbb{R} \) given by
\[ \varphi(\mathbf{v}) = x_1 + x_2 + \cdots + x_n \]
where \( \mathbf{v} = (x_1, x2, \cdots, x_n) \). Then \( \varphi \) is a linear functional.
The next proposition provides a more general example of a linear functional on \( \mathbb{R}^n \).
**Proposition 1.2**: Let \( a_1, a_2, \cdots, a_n \) be fixed real numbers. Let \( \varphi : \mathbb{R}^n \to \mathbb{R} \) be given by
\[ \varphi(\mathbf{v}) = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \]
where \( \mathbf{v} = (x_1, x_2, \cdots, x_n) \). Then \( \varphi \) is a linear functional.
**Project Problem 1**: Prove the above Proposition 1.2.
**Example 2**: Consider \( V = M_2(\mathbb{R}) \), where \( M_2(\mathbb{R}) \) denotes the set of all \( 2 \times 2 \) matrices with real entries.
Let \( \text{Tr} : M_2(\mathbb{R}) \to \mathbb{R} \) be given by
\[ \text{Tr} \left( \begin{matrix} a & b \\ c & d \end{matrix} \right) = a + d \]
Then Tr is a linear functional on \( M_2(\mathbb{R}) \), called the **Trace**. More generally, we can define a similar linear](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cf4b148-a9ff-447b-bf88-730c6d11afaf%2F7597ae1f-97ba-4ea2-9679-5790c1a71529%2F4ibhc2m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:# Linear Functionals
## Definition 1.1
Let \( V \) be a vector space over \( \mathbb{R} \). A **linear functional** on \( V \) is a linear map \( \varphi : V \to \mathbb{R} \).
### Examples:
**Example 1**: Consider \( V = \mathbb{R}^n \), and \( \varphi : \mathbb{R}^n \to \mathbb{R} \) given by
\[ \varphi(\mathbf{v}) = x_1 + x_2 + \cdots + x_n \]
where \( \mathbf{v} = (x_1, x2, \cdots, x_n) \). Then \( \varphi \) is a linear functional.
The next proposition provides a more general example of a linear functional on \( \mathbb{R}^n \).
**Proposition 1.2**: Let \( a_1, a_2, \cdots, a_n \) be fixed real numbers. Let \( \varphi : \mathbb{R}^n \to \mathbb{R} \) be given by
\[ \varphi(\mathbf{v}) = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \]
where \( \mathbf{v} = (x_1, x_2, \cdots, x_n) \). Then \( \varphi \) is a linear functional.
**Project Problem 1**: Prove the above Proposition 1.2.
**Example 2**: Consider \( V = M_2(\mathbb{R}) \), where \( M_2(\mathbb{R}) \) denotes the set of all \( 2 \times 2 \) matrices with real entries.
Let \( \text{Tr} : M_2(\mathbb{R}) \to \mathbb{R} \) be given by
\[ \text{Tr} \left( \begin{matrix} a & b \\ c & d \end{matrix} \right) = a + d \]
Then Tr is a linear functional on \( M_2(\mathbb{R}) \), called the **Trace**. More generally, we can define a similar linear
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