(Solved): Problem 3. Let V be the set of all ordered pairs of real numbers, and consider the following ad ...

Problem 3. Let $$V$$ be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operation on $$\mathbf{u}=\left(u_1,u_2\right)$$ and $$\mathbf{v}=\left(v_1,v_2\right)$$ $$\mathbf{u}+\mathbf{v}=\left(u_1+v_1?1,u_2+v_2?1\right),k\mathbf{u}=\left(k{u}_1,k{u}_2\right).$$ (a) Compute $$\mathbf{u}+\mathbf{v}$$ and $$k\mathbf{u}+(?\mathbf{v})$$ for $$\mathbf{u}=(1,?2),\mathbf{v}=(3,0)$$, and $$k=4$$. (b) Show that $$(0,0)?=0$$. (c) Show that $$(1,1)=0$$. (d) Show that for each $$\mathbf{u}$$ in $$V$$, there exists $$?\mathbf{u}$$ in $$V$$ such that $$\mathbf{u}+(?\mathbf{u})=0$$. (e) Determine whether $$V$$ is a vector space over $$\mathbf{R}$$. $$[25$$ marks] Problem 4. Find all values of $$??\mathbb{C}$$ such that $$?(1?2i,2?i,3+i)=(2?3i,4+i,1?i).$$ $$[25$$ marks] Problem 5. Prove that every vector space has a unique zero vector. [25 marks] Problem 6. Let $$??$$ and $$?$$ denote two distinct objects, neither of which is in $$\mathbf{R}$$. Define an addition and scalar multiplication on $$\mathbf{R}?\{?\}?\{??\}$$. Specifically, the sum and product of two real numbers is as usual, and for $$t?\mathbf{R}$$ define $$\begin{array}{c}t?=?\begin{array}{cl}??& \text{ if }t<0\\ 0& \text{ if }t=0\\ ?& \text{ if }t>0,\end{array}t(??)=?\begin{array}{cc}?& \text{ if }t<0\\ 0& \text{ if }t=0\\ ??& \text{ if }t>0,\end{array}\\ t+?=?+t=?,t+(??)=??+t=??,\\ ?+?=?,(??)+(??)=??,?+(??)=0.\end{array}$$ Determine whether $$\mathbf{R}?\{?\}?\{??\}$$ is a vector space over $$\mathbf{R}$$. [25 marks]