Please solve this ### 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,

As per the guideline I’m supposed to solve only first question from the given problems.

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

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 $ \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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