Section 1A 𝐑𝑛 and 𝐂𝑛 1. [[EF-1A-1]]  Show that 𝛼+𝛽 = 𝛽+𝛼 for all 𝛼,𝛽 ∈ 𝐂. [link](obsidian://open?vault=AxlerChat&file=Exercises%2FAll_Exercises%2F1A-1) find [GPT-4-Answers-Feb 2024](GPT-4-Answers-Feb%202024.md) 2. [[1A-2]] Show that(𝛼+𝛽)+𝜆 = 𝛼+(𝛽+𝜆) for all 𝛼,𝛽,𝜆∈𝐂. find[[]] 3. [[1A-3]]  Show that(𝛼𝛽)𝜆 = 𝛼(𝛽𝜆) for all 𝛼,𝛽,𝜆 ∈ 𝐂. 4. [[1A-4]]  Show that 𝜆(𝛼+𝛽) = 𝜆𝛼+𝜆𝛽 for all 𝜆,𝛼,𝛽∈𝐂. 5. [[1A-5]]  Show that for every 𝛼∈𝐂, there exists a unique 𝛽∈𝐂 such that 𝛼+𝛽 = 0. 6. [[1A-6]]  Show that for every 𝛼∈𝐂 with 𝛼≠0, there exists a unique 𝛽∈𝐂 such that 𝛼𝛽 = 1. 7. [[1A-7]] Show that −1+√3𝑖 2 is a cube root of 1 (meaning that its cube equals 1). 8. [[1A-8]]  Find two distinct square roots of 𝑖. 9. [[1A-9]]  Find 𝑥 ∈ 𝐑4 such that (4, −3, 1, 7) + 2𝑥 = (5, 9, −6, 8). 10. [[1A-10]]  Explain why there does not exist 𝜆 ∈ 𝐂 such that 𝜆(2 − 3𝑖, 5 + 4𝑖, −6 + 7𝑖) = (12 − 5𝑖, 7 + 22𝑖, −32 − 9𝑖). 11. [[1A-11]]  Show that (𝑥+𝑦)+𝑧 = 𝑥+(𝑦+𝑧) for all 𝑥,𝑦,𝑧 ∈ 𝐅𝑛. 12. [[1A-12]]  Show that (𝑎𝑏)𝑥 = 𝑎(𝑏𝑥) for all 𝑥∈𝐅𝑛 and all 𝑎,𝑏 ∈ 𝐅. 13. [[1A-13]]  Show that 1𝑥=𝑥 for all 𝑥∈𝐅𝑛. 14. [[1A-14]]  Show that 𝜆(𝑥+𝑦) = 𝜆𝑥+𝜆𝑦 for all 𝜆 ∈ 𝐅 and all 𝑥,𝑦 ∈ 𝐅𝑛. 15. [[1A-15]]  Show that (𝑎+𝑏)𝑥 = 𝑎𝑥+𝑏𝑥 for all 𝑎,𝑏 ∈ 𝐅 and all 𝑥 ∈ 𝐅𝑛. --- $ \begin{align*} &\text{Exercises 1A }\\ 1. & \text{ Show that } \alpha + \beta = \beta + \alpha \text{ for all } \alpha, \beta \in \mathbb{C}.\quad \text{} \\\\ 2. & \text{ Show that } (\alpha + \beta) + \lambda = \alpha + (\beta + \lambda) \text{ for all } \alpha, \beta, \lambda \in \mathbb{C}. \\ 3. & \text{ Show that } (\alpha \beta) \lambda = \alpha (\beta \lambda) \text{ for all } \alpha, \beta, \lambda \in \mathbb{C}. \\ 4. & \text{ Show that } \lambda (\alpha + \beta) = \lambda \alpha + \lambda \beta \text{ for all } \lambda, \alpha, \beta \in \mathbb{C}. \\ 5. & \text{ Show that for every } \alpha \in \mathbb{C}, \text{ there exists a unique } \beta \in \mathbb{C} \text{ such that } \alpha + \beta = 0. \\ 6. & \text{ Show that for every } \alpha \in \mathbb{C} \text{ with } \alpha \neq 0, \text{ there exists a unique } \beta \in \mathbb{C} \\ & \text{ such that } \alpha \beta = 1. \\ 7. & \text{ Show that } -1 + \frac{\sqrt{3}i}{2} \text{ is a cube root of } 1 \text{ (meaning that its cube equals } 1). \\ 8. & \text{ Find two distinct square roots of } i. \\ 9. & \text{ Find } x \in \mathbb{R}^4 \text{ such that } (4, -3, 1, 7) + 2x = (5, 9, -6, 8). \\ 10. & \text{ Explain why there does not exist } \lambda \in \mathbb{C} \text{ such that } \\ & \lambda(2 - 3i, 5 + 4i, -6 + 7i) = (12 - 5i, 7 + 22i, -32 - 9i). \\ 11. & \text{ Show that } (x + y) + z = x + (y + z) \text{ for all } x, y, z \in \mathbb{F}^n. \\ 12. & \text{ Show that } (ab)x = a(bx) \text{ for all } x \in \mathbb{F}^n \text{ and all } a, b \in \mathbb{F}. \\ 13. & \text{ Show that } 1x = x \text{ for all } x \in \mathbb{F}^n. \\ 14. & \text{ Show that } \lambda(x + y) = \lambda x + \lambda y \text{ for all } \lambda \in \mathbb{F} \text{ and all } x, y \in \mathbb{F}^n. \\ 15. & \text{ Show that } (a + b)x = ax + bx \text{ for all } a, b \in \mathbb{F} \text{ and all } x \in \mathbb{F}^n. \end{align*} $