1. Find three distinct non-zero vectors $\displaystyle u,v,w $ in $\displaystyle \mathbb{R}^{3} $ such that $\displaystyle \text{span}( \{u,v \}) = \text{span}( \{v,w \}) = \text{span}( \{u,v,w \}) $ but such that $\displaystyle \text{span}(\{u,w \}) \neq \text{span}(\{u,v,w \}) $.

Consider $\displaystyle u = (1,0,0) $, $\displaystyle v = (0,1,0) $, and $\displaystyle w = (1/2,0,0) $. Then $\displaystyle \text{span}( \{u,v \}) = \text{span}( \{v,w \}) = \text{span}( \{u,v,w \}) $. But $\displaystyle \text{span}(\{u,w \}) \neq \text{span}(\{u,v,w \}) $.

2. Find a basis for $\displaystyle M_{2 \times 2}^{0} (\mathbb{R}) $, the vector space of $\displaystyle 2 \times 2 $ matrices with trace zero. Why is this set a basis?

So we want to find a spanning set that is linearly independent. Consider $\displaystyle B = \left \{ \begin{bmatrix} 1 & a \\ b & -1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \right \} $ where $\displaystyle a $ and $\displaystyle b $ can be chosen freely. This is a basis because it is a linearly independent set that spans $\displaystyle M_{2 \times 2}^{0}(\mathbb{R}) $.

Is this correct?