(Solved): Consider the following set \( U \), where addition and scalar multiplication are defined as follow ...

Consider the following set \( U \), where addition and scalar multiplication are defined as follows: \[ U=\left\{\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] \in \mathbb{R}^{2}: x_{1}, x_{2}>0\right\} \quad\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]+\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right]=\left[\begin{array}{l} x_{1} y_{1} \\ x_{2} y_{2} \end{array}\right] \quad a \cdot\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{l} \left(x_{1}\right)^{a} \\ \left(x_{2}\right)^{a} \end{array}\right] \] a) Is the set \( U \) a vector space over \( \mathbb{R} \) with the above operations? Carefully justify your answer by checking each vector space axiom. If an axiom holds, give a proof; otherwise, explain why the axiom fails. b) Is the set \( U \) as subspace of \( \mathbb{R}^{2} \) ? Fully justify your answer.