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Problem 49: The addition and scalar multiplication operations as follows:
$(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)$,
$k(x_1, y_1) = (kx_1, ky_1)$.

Problem 49: We have defined the set $\R^2 = \{(x, y) : x, y \in \R\}$, together with the addition and scalar multiplication operations as follows:
$(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)$,
$k(x_1, y_1) = (kx_1, ky_1)$. Give a complete verification that each of the vector space axioms is satisfied.

Solution:
The Axioms of a Vector Space
The following properties must hold for all $ u,v,w \in \V$ and $a,b \in \R$:
Closure Properties
$(A1) u+v \in \V$.
$(A2) av \in \V$.
Properties of Addition
$(A3) u+v=v+u$.
$(A4) u+(v+w)=(u+v)+w$.
$(A5)$ There is an element $\mathbf{0} \in \V$ such that $\mathbf{0}+v=v$ for all $v \in \V$.
$(A6)$ Given an element $v \in \V$, there is an element $−v \in \V$ such that $v+(−v)=\mathbf{0}$.
Properties of Scalar Multiplication
$(A7) a(bv)=(ab)v$.
$(A8) a(u+v)=au+av$.
$(A9) (a+b)v=av+bv$.
$(A10) 1v=v$ for all $v \in \V$.

Prove A1:
Let $u,v \in V$
\begin{align*} u+v = (u_1,u_2)+(v_1,v_2) \\ = (u_1 + v_1, u_2 + v_2) \in \V \end{align*}

Prove A2:
Let $r \in \R$ and $u \in V$
\begin{align*} ru = r(u_1,u_2) \\ = (ru_1, ru_2) \in \V \end{align*}

Prove A3:
Let $u,v \in V$
\begin{align*} u+v = (u_1,u_2)+(v_1,v_2) \\ = (u_1 + v_1, u_2 + v_2)\\ = (v_1 + u_1, v_2 + u_2)\\ = v+u \end{align*}

Prove A4:
Let $u,v,w \in V$
\begin{align*} \left[u+v\right]+w = \left[(u_1 + v_1, u_2 + v_2)\right]+(w_1,w_2) \\ = (\left[u_1 + v_1\right]+w_1, \left[u_2 + v_2\right]+w_2)\\ = (u_1 + \left[v_1+w_1\right], u_2 +\left[ v_2+w_2\right])\\ = (u_1,u_2) + \left[(v_1+w_1, v_2+w_2)\right]\\ = u+ \left[v+w\right]].\\ \end{align*}

Prove A5:
Let $v\in V$ and $0=(0,0) \in \V$
\begin{align*} v+0 = (v_1,v_2)+(0,0)\\ = (v_1+0,v_2+0) \\ = (v_1,v_2)\\ = v. \end{align*}

Prove A6:
Let $u=(a,b)$ then $-u = (-a,-b), a,b \in \R$
\begin{align*} u+-u = (a,b)+(-a,-b) \\ = (a-a,b-b) \\ = (0,0) \end{align*}

Prove A7:
Let $v \in \V$ and $r,s \in \R$
\begin{align*} (rs)v = (rs)(v_1,v_2) \\ = ((rs)v_1,(rs)v_2) \\ = (rsv_1,rsv_2) \\ = (r(sv_1),r(sv_2)) \\ = r(sv_1,sv_2)\\ = r(s(v_1,v_2))\\ = r(sv) \end{align*}

Prove A8:
Let $u,v \in \V$ and $r \in \R$
\begin{align*} r(u+v) = r((u_1,u_2)+(v_1,v_2))\\ = r(u_1+v_1,u_2+v_2) \\ = (r(u_1+v_1),r(u_2+v_2))\\ = (ru_1+rv_1,ru_2+rv_2)\\ = (ru_1,ru_2)+(rv_1,rv_2)\\ = r(u_1,u_2)+r(v_1,v_2)\\ = ru+rv. \end{align*}

Prove A9:
Let $u\in \V$ and $r,s \in \R$
\begin{align*} (r+s)u = (r+s)(u_1,u_2)\\ = ((r+s)u_1,(r+s)u_2)\\ = (ru_1+su_1,ru_2+su_2)\\ = (ru_1,ru_2)+(su_1,su_2)\\ = r(u_1,u_2)+s(u_1,u_2)\\ = ru+su \end{align*}

Prove A10:
Let $v\in \V$ and $1 \in \R$
\begin{align*} 1\cdot v = 1(v_1,v_2)\\ = (1\cdot v_1, 1\cdot v_2)\\ = (v_1,v_2)\\ = v \end{align*}

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