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 q(0) = α₁x₁ + x2x2 + ... + anxn, where v = (x₁,x2,...,xn). Then q is a linear functional. Project Problem 1: Prove the above Proposition 1.2
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 q(0) = α₁x₁ + x2x2 + ... + anxn, where v = (x₁,x2,...,xn). Then q is a linear functional. Project Problem 1: Prove the above Proposition 1.2
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### 1. 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:**
1. **Example 1:** Consider \( V = \mathbb{R}^n \), and \( \varphi : \mathbb{R}^n \to \mathbb{R} \) given by \( \varphi(\vec{v}) = x_1 + \cdots + x_n \), where \( \vec{v} = (x_1, x_2, \cdots, x_n) \). 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 \( \varphi(\vec{v}) = a_1x_1 + a_2x_2 + \cdots + a_n x_n \), where \( \vec{v} = (x_1, x_2, \cdots, x_n) \). Then \( \varphi \) is a linear functional.
- **Project Problem 1:** Prove the above Proposition 1.2.
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 \( \operatorname{Tr} : M_2(\mathbb{R}) \to \mathbb{R} \) be given by \( \operatorname{Tr}\left[\begin{matrix} a & b \\ c & d \end{matrix}\right] = a + d \).
Then \( \operatorname{Tr} \) is a linear functional on \( M_2(\mathbb{R}) \), called the Trace. More generally,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cf4b148-a9ff-447b-bf88-730c6d11afaf%2Fb2dfad2a-9e51-4acb-97c6-ae308e983d67%2Fmxpwzx6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### 1. 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:**
1. **Example 1:** Consider \( V = \mathbb{R}^n \), and \( \varphi : \mathbb{R}^n \to \mathbb{R} \) given by \( \varphi(\vec{v}) = x_1 + \cdots + x_n \), where \( \vec{v} = (x_1, x_2, \cdots, x_n) \). 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 \( \varphi(\vec{v}) = a_1x_1 + a_2x_2 + \cdots + a_n x_n \), where \( \vec{v} = (x_1, x_2, \cdots, x_n) \). Then \( \varphi \) is a linear functional.
- **Project Problem 1:** Prove the above Proposition 1.2.
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 \( \operatorname{Tr} : M_2(\mathbb{R}) \to \mathbb{R} \) be given by \( \operatorname{Tr}\left[\begin{matrix} a & b \\ c & d \end{matrix}\right] = a + d \).
Then \( \operatorname{Tr} \) is a linear functional on \( M_2(\mathbb{R}) \), called the Trace. More generally,
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