(Solved): AAT Exercise 12022/23 1. Let V=R2 with the standard vector addition and scalar multiplication defi ...

AAT Exercise $$12022/23$$ 1. Let $$V={\mathbb{R}}^2$$ with the standard vector addition and scalar multiplication defined as $$c\left(x_1,x_2\right)=$$ $$\left(x_1,c{x}_2\right)$$. Decide whether $$V$$ is a vector space over $$\mathbb{R}$$. 2. Suppose that the set $$V$$ is the set of positive real numbers (i.e $$x>0)$$ with addtion and scalar multiplication defined as follows. $$?x,y?V$$ and $$c?\mathbb{R},x+y=xy$$ and $$cx=x^c$$. Show that $$V$$ under this addition and scalar multiplication is a vector space. 3. Give a $$1?1$$ correspondence between the vector space $${\mathbb{Q}}^4$$ over $$\mathbb{Q}$$ and the vector space $$M_{2\times 2}(\mathbb{Q})$$. (Hint: Find the basis for the given vector spaces and give a bijection from one basis to the other). 4. In each case decide whether the given formulas defines a norm on $${\mathbb{R}}^3$$. If so, provide a proof; if not give reasons. (i) $$?\underline{x}=\left(x_1,x_2,x_3\right)?{\mathbb{R}}^3?\underline{x}?=\max \left\{?x_1?,?x_2?,?x_3?\right\}$$. (ii) $$?\underline{x}=\left(x_1,x_2,x_3\right)?{\mathbb{R}}^3?\underline{x}??=?x_1+x_2+x_3?$$. (iii) $$?\underline{x}=\left(x_1,x_2,x_3\right)?{\mathbb{R}}^3?\underline{x}?=a\left(?x_1?+?x_2?+?x_3?\right)?a?\mathbb{R},a>0$$. (iv) $$?\underline{x}=\left(x_1,x_2,x_3\right)?{\mathbb{R}}^3?\underline{x}?=\frac{?\underline{x}?}{1+?\underline{x}?}$$ where $$?\underline{x}?$$ is a norm on $${\mathbb{R}}^3$$.