Section 1B Definition of Vector Space 17 1. [[1B.1]]  Prove that−(−𝑣)=𝑣 for every 𝑣 ∈ 𝑉. 2. [[1B.2]]  Suppose𝑎∈𝐅,𝑣∈𝑉,and𝑎𝑣=0.Provethat𝑎=0or𝑣=0. 3. [[1B.3]]  Suppose 𝑣, 𝑤 ∈ 𝑉. Explain why there exists a unique 𝑥 ∈ 𝑉 such that 𝑣 + 3𝑥 = 𝑤. 4. [[1B.4 ]] The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in the definition of a vector space (1.20). Which one? 5. [[1B.5]]  Show that in the definition of a vector space (1.20), the additive inverse condition can be replaced with the condition that 0𝑣 = 0 for all 𝑣 ∈ 𝑉. Here the 0 on the left side is the number 0, and the 0 on the right side is the additive identity of 𝑉. The phrase a “condition can be replaced” in a definition means that the collection of objects satisfying the definition is unchanged if the original condition is replaced with the new condition. Section 1B Definition of Vector Space 17 [[1B.6]] Let ∞ and −∞ denote two distinct objects, neither of which is in $\mathbb{R}$. Define an addition and scalar multiplication on $\mathbb{R}$ ∪ {∞, −∞} as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and for 𝑡 ∈ $mathbb{R}$ define and (−∞)if 𝑡 < 0, 𝑡∞ = 0 if 𝑡 = 0, ∞ if 𝑡 > 0, 𝑡 + ∞ = ∞ + 𝑡 = ∞ + ∞ = ∞, 𝑡+(−∞) = (−∞)+𝑡 = (−∞)+(−∞) = −∞, {⎨0 {∞ {−∞ ⎧{−∞ if 𝑡 < 0, if 𝑡 = 0,⎧{∞ 𝑡(−∞) = {⎨0 if 𝑡 > 0, ⎩⎩∞+(−∞) = (−∞)+∞ = 0. With these operations of addition and scalar multiplication, is $mathbb{R}$ ∪ {∞, −∞} a vector space over $mathbb{R}$? Explain. 1. [[1B.7]]  Suppose 𝑆 is a nonempty set. Let 𝑉𝑆 denote the set of functions from 𝑆 to 𝑉. Define a natural addition and scalar multiplication on 𝑉𝑆, and show that 𝑉𝑆 is a vector space with these definitions. 2. [[1B.8]]  Suppose 𝑉 is a real vector space. - The complexification of 𝑉, denoted by 𝑉𝐂 , equals 𝑉 × 𝑉. An element of $V_C$ is an ordered pair(𝑢,𝑣), where 𝑢,𝑣 ∈ 𝑉, but we write this as 𝑢+𝑖𝑣. - Addition on $V_C$ is defined by (𝑢1 +𝑖𝑣1)+(𝑢2 +𝑖𝑣2) = (𝑢1 +𝑢2)+𝑖(𝑣1 +𝑣2) forall𝑢1,𝑣1,𝑢2,𝑣2 ∈𝑉. - Complex scalar multiplication on $V_C$ is defined by (𝑎+𝑏𝑖)(𝑢+𝑖𝑣) = (𝑎𝑢−𝑏𝑣)+𝑖(𝑎𝑣+𝑏𝑢) for all 𝑎, 𝑏 ∈ $mathbb{R}$ and all 𝑢, 𝑣 ∈ 𝑉. Prove that with the definitions of addition and scalar multiplication as above, $V_C$ is a complex vector space. Think of 𝑉 as a subset of $V_C$ by identifying 𝑢 ∈ 𝑉 with 𝑢 + 𝑖0. The construction of $V_C$ from 𝑉 can then be thought of as generalizing the construction of 𝐂𝑛 from $\mathbb{R}^𝑛$. --- ## MathJax Here is the provided text formatted for display in MathJax: $ \begin{aligned} \text{Exercises 1B} \\ \hline\\ 1. & \text{ Prove that } -(-\mathbf{v}) = \mathbf{v} \text{ for every } \mathbf{v} \in \mathbf{V}. \\\\ 2. & \text{ Suppose } a \in \mathbb{F}, \mathbf{v} \in \mathbf{V}, \text{ and } a\mathbf{v} = 0. \text{ Prove that } a = 0 \text{ or } \mathbf{v} = 0. \\\\ 3. & \text{ Suppose } \mathbf{v}, \mathbf{w} \in \mathbf{V}. \text{ Explain why there exists a unique } \mathbf{x} \in \mathbf{V} \\&\text{ such that } \mathbf{v} + 3\mathbf{x} = \mathbf{w}. \\\\ 4. & \text{ The empty set is not a vector space.} \text{ The empty set fails to satisfy} \\&\text{ only one of the requirements listed in the definition of a vector space(1.20).} \\&\text{ Which one?} \\\\ 5. & \text{ Show that in the definition } \\&\text{of a vector space (1.20), }\text{the additive inverse condition can be replaced} \\&\text{ with the condition that } 0\mathbf{v} = 0 \\& \text{ for all } \mathbf{v} \in \mathbf{V}. \\\\ & \text{ Here the } 0 \text{ on the left side is the number } 0, \\&\text{ and the } 0 \text{ on the right side is the additive identity of } \mathbf{V}. \\ & \text{ The phrase a "condition can be replaced" } \\&\text{in a definition } \\&\text{means that the collection of objects satisfying the definition } \\&\text{is unchanged if the original condition is replaced } \\&\text{with the new condition.} \\\\ 6. & \text{ Let } \infty \text{ and } -\infty \text{ denote two distinct objects, neither of which is in } \mathbb{R}. \\& \text{ Define an addition and scalar multiplication on } \mathbb{R} \cup \{\infty, -\infty\}\\&\text{ as you could guess from the notation. }\\&\text{Specifically, the sum and product of two real numbers is }\\&\text{as usual, and for } t \in \mathbb{R} \\& \text{ define} \\\\ &t\infty = \begin{cases} -\infty &\text{if } t < 0, \\ 0 &\text{if } t = 0\\ \infty & \text{if } t > 0, \\ \end{cases} \\ &\text{and}\\ &t(-\infty) = \begin{cases} \infty&\text{if } t < 0,\\ 0 &\text{if } t = 0\\ -\infty & \text{if } t < 0, \\ \end{cases} \\ &\text{and}\\\\ %&\begin{cases} &t + \infty = \infty + t = \infty + \infty = \infty, \\ &t + (-\infty) = (-\infty) + t = (-\infty) + (-\infty) = -\infty, \\ &\infty + (-\infty) = (-\infty) + \infty = 0. \\\\ &\text{and}\\\\ & \infty + t = t + \infty = \infty + \infty = \infty, \\ & t + (-\infty) = (-\infty) + t = (-\infty) + (-\infty) = -\infty, \\ & \infty + (-\infty) = (-\infty) + \infty = 0. \\\\ & \text{ With these operations of addition and scalar multiplication, is } \mathbb{R} \cup \{\infty, -\infty\}\\& \text{ a vector space over } \mathbb{R}? \text{ Explain.} \\\\ 7. & \text{ Suppose } S \text{ is a nonempty set. Let } V_S \text{ denote the set of functions from } S \text{ to } V.\\& \text{ Define a natural addition and scalar multiplication on } V_S, \text{ and show that } V_S\\& \text{ is a vector space with these definitions.} \\\\ 8. & \text{ Suppose } V \text{ is a real vector space.} \\\\ & \bullet \text{ The complexification of } V, \text{ denoted by } V_{\mathbb{C}}, \text{ equals } V \times V. \text{ An element of } \\&V_{\mathbb{C}} \text{ is an ordered pair } (u,v), \text{ where } u,v \in V, \text{ but we write this as } u + iv. \\ & \bullet \text{ Addition on } V_{\mathbb{C}} \text{ is defined by} \\ & (u_1 + iv_1) + (u_2 + iv_2) = (u_1 + u_2) + i(v_1 + v_2) \text{ for all } u_1,v_1,u_2,v_2 \in V. \\ & \bullet \text{ Complex scalar multiplication on } V_{\mathbb{C}} \text{ is defined by} \\ & (a + bi)(u + iv) = (au - bv) + i(av + bu) \\ & \text{ for all } a, b \in \mathbb{R} \text{ and all } u, v \in V. \\ & \text{ Prove that with the definitions of}\text{ addition and scalar multiplication as above, }\\& V_{\mathbb{C}} \text{ is a complex vector space.} \\\\ & \text{ Think of } V \text{ as a subset of } V_{\mathbb{C}} \text{ by identifying } u \in V \text{ with } u + i0.\\& \text{ The construction of } V_{\mathbb{C}} \text{ from } V \text{ can then be thought of as }\\&\text{generalizing the construction of } \mathbb{C}^n \text{ from } \mathbb{R}^n. \\ \end{aligned} $